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标题: 数学专业英语[1]－The Real Number System [打印本页]

作者: hehe123 时间: 2004-11-27 12:21
标题: 数学专业英语[1]－The Real Number System
数学专业英语－The Real Number System

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The Real-Number System , I7 T3 [ ^8 t8 M4 F

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The real-number system is collection of mathematical objects, called real number, which acquire mathematical life by virtue fundamental principles, or rules, that we adopt. The situation is somewhat similar to a game, like chess, for example. The chess system, or game, is a collection of objects, called chess pieces, which acquire life by virtue of the rules of the game, that is, the principles that are adopted to define allowable moves for the pieces and the way in which they may interact.

Our working experience with numbers has provided us all with some familiarity with the principles that govern the real-number system. However, to establish a common ground of understanding and avoid certain errors that have become very common, we shall explicitly state and illustrate many of these principles.

The real-number system includes such numbers as –27,-2,2/3,… It is worthy of note that positive numbers, 1/2, 1, for examples, are sometimes expressed as +(1/2), +1. The plus sign, “+”, used here does not express the operation of addition, but is rather part of the symbolism for the numbers themselves. Similarly, the minus sign, “-“, used in expressing such numbers as -(1/2), -1, is part of the symbolism for these numbers.

Within the real number system, numbers of various kinds are identified and named. The numbers 1, 2, 3, 4,… which are used in the counting process, are called natural numbers. The natural numbers, together with–1,-2,-3,-4,…and zero, are called integers. Since 1,2,3,4,…are greater than 0, they are also called positive integers; -1,-2,-3,-4,…are less than 0, and for this reason are called negative integers. A real number is said to be a rational number if it can be expressed as the ratio of two integers, where the denominator is not zero. The integers are included among the rational numbers since any integer can be expressed as the ratio of the integer itself and one. A real number that cannot be expressed as the ratio of two integers is said to be an irrational number.

One of the basic properties of the real-number system is that any two real numbers can be compared for size. If a and b are real numbers, we write a<b to signify that a is less than b. Another way of saying the same thing is to write b>a, which is read “b is greater than a “.

Geometrically, real numbers are identified with points on a straight line. We choose a straight line, and an initial point f reference called the origin. To the origin we assign the number zero. By marking off the unit of length in both directions from the origin, we assign positive integers to marked-off points in one direction (by convention, to the right of the origin ) and negative integers to marked-off point in the other direction. By following through in terms of the chosen unit of length, a real number is attached to one point on the number line, and each point on the number line has attached to it one number.

Geometrically, in terms of our number line, to say that a<b is to say that a is to the left of b; b>a means that b is to the right of a.

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Properties of Addition and Multiplication

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Addition and multiplication are primary operations on real numbers. Most, if not all, of the basic properties of these operations are familiar to us from experience.

(a) Closure property of addition and multiplication.

Whenever two real numbers are added or multiplied, we obtain a real number as the result. That is, performing the operations of addition and multiplication leaves us within the real-number system.

(b) Commutative property of addition and multiplication.

The order in which two real numbers are added or multiplied does not affect the result obtained. That is, if a and b are any two real numbers, then we have (i) a+ b=b+ a and (ii) ab = ba. Such a property is called a commutative property. Thus, addition and multiplication of real numbers are commutative operations.

Parentheses, brackets, and the like, we recall, are used in algebra to group together whatever terms are within them. Thus 2+(3+4) means that 2 is to be added to the sum of 3 and 4 yielding 2+7 =9 whereas (2+3)+4 means the sum of 2 and 3 is to be added to 4 yielding also 9. Similarly, 2•(3•4) yields 2•(12)=24 whereas (2•3) •4 yields the same end result by the route 6•4=24 . That such is the case in general is the content of the associative property of addition and multiplication of real numbers.

(d) Distributive property of multiplication over addition.

We know that 2•(3•4)=2•7=14 and that 2•3+ 2•4=14 ,thus 2•(3+4)=2•3+ 2•4. That such is the case in general for all real numbers is the content of the distributive property of multiplication over addition, more simply called the distributive property.

Substraction and Division

The numbers zero and one. The following are the basic properties of the numbers zero and one.

(a) There is a unique real number, called zero and denoted by 0, with the property that a+0=0+a, where a is any real number.

There is a unique real number, different from zero, called one and denoted by 1, with the property that a•1=1•a=a, where a is any real number.

(b) If a is any real number, then there is a unique real number x, called the additive inverse of a , or negative of a, with the property that a+ x = x+ a .If a is any nonzero real number, then there is a unique real number y, called the multiplicative inverse of a, or reciprocal of a, with the property that ay = ya = 1

The concept of the negative of a number should not be confused with the concept of a negative number; they are not the same. ”Negative of“ means additive inverse of “. On the other hand, a “negative number” is a number that is less than zero.

The multiplicative inverse of a is often represented by the symbol 1/a or a^-1. Note that since the product of any number y and 0 is 0, 0 cannot have a multiplicative inverse. Thus 1/0 does not exist.

Now substraction is defined in terms of addition in the following way. . p) p, \4 j9 v% B! Q! x

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If a and b are any two real numbers, then the difference a-b is defined by a- b= c where c is such that b+ c=a or c= a+(-b). That is, to substract b from a means to add the negative of b (additive inverse of b) to a.

Division is defined in terms of multiplication in the following way.

If a and b are any real numbers, where b≠0, then a+ b is defined by a +b= a•(1/b) =a•b^-1. That is, to divide a by b means to multiply a by the multiplicative inverse ( reciprocal)of b. The quotient a +b is also expressed by the fraction symbol a/b.

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作者: hehe123 时间: 2004-11-27 12:22

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Vocabulary X8 | I0 L# G

real number 实数

negative 负的

the real number system 实数系

rational number 有理数

collection 集体,总体

ratio 比,比率

object 对象,目的

denominator 分母

principle 原理,规则

numerator 分子

adopt 采用

irrational number 无理数

define 定义(动词)

signify 表示

definition 定义(名词)

geometrical 几何的

establish 建立

straight line 直线

explicit 清晰的,明显的

initial point 初始点

illustrate 说明

point of reference 参考点

positive 正的

origin 原点

express 表达

assign 指定

plus 加

unit 单位

sign 记号,符号,正负号

property 性质

operation 运算,操作

closure property 封闭性质

addition 加法

commutative 交换的

multiplication 乘法

associative 结合的

substraction 减法

parentheses 圆括号

division 除法

brackets 括号

sum 和,总数

algebra 代数

procuct 乘积

yield 产生

difference 差,差分

term 术语,项

quotient 商

distributive 分配的

symbolism 符号系统

unique 唯一的

minus 减

additive inverse 加法逆运算

identify 使同一

multiplicative inverse 乘法逆运算

count 计数

reciprocal 倒数,互逆

natural number 自然数

concept 概念

zero 零

fraction 分数

integer 整数

arithmetic 算术的

greater than 大于

solution 解,解法

less than 小于

even 偶的

be equal to 等于

odd 奇的

arbitrary 任意的

square 平方

absolute value 绝对值

square root 平方根

cube 立方

induction 归纳法

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作者: hehe123 时间: 2004-11-27 12:22

Note

1. Our working experience with numbers has provided us all with some familiarity with the principles that govern the real-number system.

意思是:我们对数的实际工作经验使我们大家对支配着实数系的各原则早已有了某些熟悉,这里working作”实际工作的”解,govern作”支配”解.

2.The plus sign,”+”, used here not express the operation of addition, but is rather part of the symbolism for the numbers themselves.

意思是:这里的正符号”+”不是表示加法运算,而是数本身的符号系统的一部分.

3. A real number is said to be a rational number if it can be expressed as the ratio of two integers, where the denominator is not zero.

这是定义数学术语的一种形式.下面是另一种定义数学术语的形式.

A matrix is called a square matrix if the number of its rows equals the number of its columns.

这里is called与is said to be 可以互用,注意is called后面一般不加to be而is said后面一般要加.

4. A real number that cannot be expressed as the ratio of two integers is said to be an irrational number.

与注3比较,这里用定语从句界定术语.

5. There is a unique real number, called zero and denoted by 0, with the property that a+0=0+a, where a is any real number.

意思是:存在唯一的一个实数,叫做零并记为0,具有性质a+0=0+a,这里(其中)a是任一实数.

1) 这里called和denoted都是过去分词,与后面的词组成分词短语,修饰number.

2) with the property是前置短语,修饰number.

3) 注意本句和注3.中where的用法,一般当需要附加说明句子中某一对象时可用此结构.

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作者: hehe123 时间: 2004-11-27 12:24

Exercise

I. Turn the following arithmetic expressions into English:

i) 3+(-2)=1 ii) 2+3(-4)=-10

iii) = -5 iv) =3

v)2/5-1/6=7/30

II. Fill in each blank the missing mathematical term to mark the following sentences complete.

i) The of two real numbers of unlike signs is negative.

ii) An integer n is called if n=2m for some integer m.

iii) An solution to the equation xⁿ=c is called the n is of c.

iv) If x is a real number, then the of x is a nonnegative real number denoted by |x| and defined as follows

x, if x 0

|x|= -x, if x <0

III. Translate the following exercises into Chinese:

i) If x is an arbitrary real number, prove that there is exactly one integer n such that x<n<x+1.

ii) Prove that there is no rational number whose square in 2.

iii) Given positive real numbers a1,a2,a3,…such that an<ca_n-1 for all n>2, where c is a fixed positive number, use induction to prove that an<c^n-1a₁, for all n>1.

iv) Determine all positive integers n for which 2ⁿ<n!

Ⅳ Translate the following passage into Chinese:

There are many ways to introduce the real number system. One popular method is to begin with the positive integers 1,2,3,…and use them as building blocks to construct a more comprehensive system having the properties desired. Briefly, the idea of this method is to take the positive integers as undefined concepts, state some axioms concerning them, and them use the positive integers to build a larger system consisting of the positive rational numbers. The positive irrational numbers, in turn, may then be used as basis for constructing the positive irrational numbers. The final step is the introduction of the negative numbers and zero. The most difficult part of the whole process is the transition from the rational numbers to the irrational num

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Ⅴ. Translate the following theorems into English:

1. 定理A: 每一非负数有唯一一个非负平方根.

2. 定理B: 若x>0, y是任意一实数,则存在一正整数n使得nx > y.

Ⅵ. 1. Try to show the structure of the set of real numbers graphically.

2. List and state the laws that operations of addition and multiplication of real numbers obey.

作者: hehe123 时间: 2004-11-27 12:26
标题: 数学专业英语[2]－Basic Concepts of the Theory of Sets
数学专业英语－Basic Concepts of the Theory of Sets

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In discussing any branch of mathematics, be it analysis, algebra, or geometry, it is helpful to use the notation and terminology of set theory. This subject, which was developed by Boole and Cantor in the latter part of the 19^thcentury, has had a profound influence on the development of mathematics in the 20^th century. It has unified many seemingly disconnected ideas and has helped to reduce many mathematical concepts to their logical foundations in an elegant and systematic way. A thorough treatment of theory of sets would require a lengthy discussion which we regard as outside the scope of this book. Fortunately, the basic noticns are few in number, and it is possible to develop a working knowledge of the methods and ideas of set theory through an informal discussion . Actually, we shall discuss not so much a new theory as an agreement about the precise terminology that we wish to apply to more or less familiar ideas.

In mathematics, the word “set” is used to represent a collection of objects viewed as a single entity

The collections called to mind by such nouns as “flock”, “tribe”, ‘crowd”, “team’, are all examples of sets, The individual objects in the collection are called elements or members of the set, and they are said to belong to or to be contained in the set. The set in turn ,is said to contain or be composed of its elements.

We shall be interested primarily in sets of mathematical objects: sets of numbers, sets of curves, sets of geometric figures, and so on. In many applications it is convenient to deal with sets in which nothing special is assumed about the nature of the individual objects in the collection. These are called abstract sets. Abstract set theory has been developed to deal with such collections of arbitrary objects, and from this generality the theory derives its power.

NOTATIONS. Sets usually are denoted by capital letters: A,B,C,….X,Y,Z ; elements are designated by lower-case letters: a, b, c,….x, y, z. We use the special notation

X∈S

To mean that “x is an element of S “or” x belongs to S”. If x does not belong to S, we write x∈S. When convenient ,we shall designate sets by displaying the elements in braces; for example ，the set of positive even integers less than 10 is denoted by the symbol{2,4,6,8}whereas the set of all positive even integers is displayed as {2,4,6,…},the dots taking the place of “and so on”.

The first basic concept that relates one set to another is equality of sets:

DEFINITION OF SET EQUALITY Two sets A and B are said to be equal(or identical)if they consist of exactly the same elements, in which case we write A=B. If one of the sets contains an element not in the other ,we say the sets are unequal and we write A≠B.

SUBSETS. From a given set S we may form new sets, called subsets of S. For example, the set consisting of those positive integers less than 10 which are divisible by 4(the set{4,8})is a subset of the set of all even integers less than 10.In general, we have the following definition.

DEFINITION OF A SUBSET.A set A is said to be a subset of a set B, and we write

A B

Whenever every element of A also belongs to B. We also say that A is contained in B or B contains A. The relation is referred to as set inclusion.

The statement A B does not rule out the possibility that B A. In fact, we may have both A B and B A, but this happens only if A and B have the same elements. In other words, A=B if and only if A B and B A .

This theorem is an immediate consequence of the foregoing definitions of equality and inclusion. If A B but A≠B, then we say that A is a proper subset of B: we indicate this by writing A B.

In all our applications of set theory, we have a fixed set S given in advance, and we are concerned only with subsets of this given set. The underlying set S may vary from one application to another; it will be referred to as the universal set of each particular discourse.

The notation

{X∣X∈S. and X satisfies P}

will designate the set of all elements X in S which satisfy the property P. When the universal set to which we are referring id understood, we omit the reference to S and we simply write{X∣X satisfies P}.This is read “the set of all x such that x satisfies p.” Sets designated in this way are said to be described by a defining property For example, the set of all positive real numbers could be designated as {X∣X>0};the universal set S in this case is understood to be the set of all real numbers. Of course, the letter x is a dummy and may be replaced by any other convenient symbol. Thus we may write

{x∣x>0}={y∣y>0}={t∣t>0}

and so on .

It is possible for a set to contain no elements whatever. This set is called the empty set or the void set, and will be denoted by the symbolφ.We will consider φto be a subset of every set. Some people find it helpful to think of a set as analogous to a container(such as a bag or a box)containing certain objects, its elements. The empty set is then analogous to an empty container.

To avoid logical difficulties, we must distinguish between the element x and the set {x} whose only element is x ,(A box with a hat in it is conceptually distinct from the hat itself.)In particular, the empty setφis not the same as the set {φ}.In fact, the empty set φcontains no elements whereas the set {φ} has one element φ(A box which contains an empty box is not empty).Sets consisting of exactly one element are sometimes called one-element sets.

UNIONS,INTERSECTIONS, COMPLEMENTS. From two given sets A and B, we can form a new set called the union of A and B. This new set is denoted by the symbol

A∪B(read: “A union B”)

And is defined as the set of those elements which are in A, in B, or in both. That is to say, A∪B is the set of all elements which belong to at least one of the sets A,B.

Similarly, the intersection of A and B, denoted by

A∩B(read: “A intersection B”)

Is defined as the set of those elements common to both A and B. Two sets A and B are said to be disjoint if A∩B=φ.

If A and B are sets, the difference A-B (also called the complement of B relative to A)is defined to be the set of all elements of A which are not in B. Thus, by definition,

A- B={X|X∈A and X B}

The operations of union and intersection have many formal similarities with (as well as differences from) ordinary addition and multiplications of union and intersection, it follows that A∪B=B∪A and A∩B=B∩A. That is to say, union and intersection are commutative operations. The definitions are also phrased in such a way that the operations are associative:

(A∪B)∪C=A∪(B∪C)and(A∩B)∩C=A=∩(B∩C).

The operations of union and intersection can be extended to finite or infinite collections of sets.

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作者: hehe123 时间: 2004-11-27 12:26

Vocabulary

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Set 集合 proper subset 真子集

Set theory 集合论 universal set 泛集

Branch 分支 empty set空集

Analysis 分析 void set 空集

Geometry 几何学 union 并，并集

Notation 记号，记法 intersection交，交集

Terminology 术语，名词表 complement余，余集

Logic 逻辑 relative to相对于

Logical 逻辑的 finite有限的

Systematic 系统的 disjoint不相交

Informal 非正式的 infinite无限的

Formal正式的 cardinal number基数，纯数

Entity 实在物 ordinal number序数

Element 元素 generality一般性，通性

Abstract set 抽象集 subset子集

Designate 指定， divisible可除的

Notion 概念 set inclusion 集的包含

Braces 大括号 immediate consequence直接结果

Identical 恒同的，恒等的

作者: hehe123 时间: 2004-11-27 12:27

Notes % e- \8 ^5 Q7 f5 {

1. In discussing any branch of mathematics, be it analysis, algebra, or geometry, it is helpful to use the notation and terminology of set theory.

意思是：在讨论数学的任何分支时，无论是分析，代数或分析，利用集合论的记号和术语是有帮助的。

这一句中be it analysis, algebra, or geometry 是以be开头的状语从句，用倒装形式。类似的句子还有：

people will use the tools in further investigations, be it in mathematic, hysics , or what have you .

2. Actually, we shall discuss not so much a new theory as an agreement about the precise terminology that we wish to more or less familiar ideas.

意思是：事实上，我恩将讨论的与其说是一种新理论，不如说是关于精确术语的一种约定，我们希望将它们应用到或多或少熟悉的思想上去。

注意：not so much A as B 在这里解释为“与其说A不如说B。”类似的用法如：

This is not so much a lecture as a friendly chat.

(与其说这是演讲不如说是朋友间的交谈。)

3．Two sets A and B are said to be equal if they consist of exactly the same elements, in which case we write A=B.

数学上常常在给定了定义后，就用符号来表达。上面句子是常见句型。类似的表达法有：

A set A is said to be a subset of a set B, and we write A=B whenever every element of A also belongs to B.

This set is called the empty set or the void set, and will be denoted by the symbol Φ.

作者: hehe123 时间: 2004-11-27 12:28

Exercise 2 O- p0 F" T7 j4 e' c & @# b8 ?) |7 Z ) q4 E' F/ }/ v! a1 B) K

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ⅰ. Turn the following mathematical expressions in English:

ⅰ)x∈A∪B ⅱ)A∩B=φ

ⅲ)A={Φ} ⅳ)A={X: a<x<b}

ⅱ.Let A ={2,5,8,11,14} B={2,8,14} C={2,8}

D={5,11} E={2,8,11}

ⅰ)B,C,D and E are ____________of A.

ⅱ)C is the ______________of B and E.

ⅲ)A is the ______________of B and D.

ⅳ)The intersection of B and D is ____________

Read the text carefully and then insert the insert the correct mathematical term in each of the blanks.

ⅲ)Give the definition of each of the following:

1.A two_ element set.

2.The difference set of A and B, where A and B are sets.

ⅳ.Four statements are given below. Among them, there is one and only one statement that cannot be used to express the meaning of A∩B=ф.Point it out and give your reason.

a) The intersection of A and B is zero.

b) Set A does not intersect set B.

c) The intersection of A and B is zero.

d) Set A and set B are B are disjoint.

ⅴ.Translate the following passage into Chinese:

It was G.. Cantor who introduced the concept the concept of the set as an object of mathematical study. Cantor stated: “A set is a collection of definite, well_ distinguished Objects of out intuition or thought. These objects are called the elements of the set. “cantor introduced the notions of cardinal and ordinal number and developed what is now known as Set Theory.

ⅵ Translate the following sentences into English:

1．若集A 与集B均是集C的子集，则集A与集B的并集仍是集C的子集。

2．集A的补（余）集的补集是A 。

[此贴子已经被作者于2004-11-27 12:29:10编辑过]

作者: hehe123 时间: 2004-11-27 12:29
标题: 数学专业英语[3]－Continuous Functions of One Real Variable
数学专业英语－Continuous Functions of One Real Variable 5 r6 z$ S1 p" W! U: l% C& C/ W

This lesson deals with the concept of continuity, one of the most important and also one of the most fascinating ideas in all of mathematics. Before we give a preeise technical definition of continuity, we shall briefly discuss the concept in an informal and intuitive way to give the reader a feeling for its meaning.

Roughly speaking the situation is this: Suppose a function f has the value f ( p ) at a certain point p. Then f is said to be continuous at p if at every nearby point x the function value f ( x ) is close to f ( p ). Another way of putting it is as follows: If we let x move toward p, we want the corresponding function value f ( x ) to become arbitrarily close to f ( p ), regardless of the manner in which x approaches p. We do not want sudden jumps in the values of a continuous function.

Consider the graph of the function f defined by the equation f ( x ) = x –[ x ], where [ x ] denotes the greatest integer < x . At each integer we have what is known ad a jump discontinuity. For example, f ( 2 ) = 0 ,but as x approaches 2 from the left, f ( x ) approaches the value 1, which is not equal to f ( 2 ).Therefore we have a discontinuity at 2. Note that f ( x ) does approach f ( 2 ) if we let x approach 2 from the right, but this by itself is not enough to establish continuity at 2. In case like this, the function is called continuous from the right at 2 and discontinuous from the left at 2. Continuity at a point requires both continuity from the left and from the right.

In the early development of calculus almost all functions that were dealt with were continuous and there was no real need at that time for a penetrating look into the exact meaning of continuity. It was not until late in the 18^th century that discontinuous functions began appearing in connection with various kinds of physical problems. In particular, the work of J.B.J. Fourier(1758-1830) on the theory of heat forced mathematicians the early 19^th century to examine more carefully the exact meaning of the word “continuity”.

A satisfactory mathematical definition of continuity, expressed entirely in terms of properties of the real number system, was first formulated in 1821 by the French mathematician, Augustin-Louis Cauchy (1789-1857). His definition, which is still used today, is most easily explained in terms of the limit concept to which we turn now.

The definition of the limit of a function.

Let f be a function defined in some open interval containing a point p, although we do not insist that f be defined at the point p itself. Let A be a real number.

The equation

f ( x ) = A % s+ j* L4 O& @ . e/ p/ o4 ~" t) y- \- S8 A1 k4 Z* u: N: a

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is read “The limit of f ( x ) , as x approached p, is equal to A”, or “f ( x ) approached A as x approached p.” It is also written without the limit symbol, as follows:

f ( x )→ A as x → p 0 P. W" a/ P# J7 i

9 a- }' U8 D3 D6 @2 Q

9 i* j2 ~. Z2 G( C$ |

This symbolism is intended to convey the idea that we can make f ( x ) as close to A as we please, provided we choose x sufficiently close to p.

Our first task is to explain the meaning of these symbols entirely in terms of real numbers. We shall do this in two stages. First we introduce the concept of a neighborhood of a point, the we define limits in terms of neighborhoods.

Definition of neighborhood of a point.

Any open interval containing a point p as its midpoint is called a neighborhood of p.

NOTATION. We denote neighborhoods by N ( p ), N₁ ( p ), N₂ ( p ) etc. Since a neighborhood N ( p ) is an open interval symmetric about p, it consists of all real x satisfying p-r < x < p+r for some r > 0. The positive number r is called the radius of the neighborhood. We designate N ( p ) by N ( p; r ) if we wish to specify its radius. The inequalities p-r < x < p+r are equivalent to –r<x-p<r, and to ∣x-p∣< r. Thus N ( p; r ) consists of all points x whose distance from p is less than r.

In the next definition, we assume that A is a real number and that f is a function defined on some neighborhood of a point p (except possibly at p ) . The function may also be defined at p but this is irrelevant in the definition.

Definition of limit of a function.

The symbolism

f ( x ) = A or [ f ( x ) → A as x→ p ] 0 M) Y7 ~% K& N# O

+ D6 @: \ Q- j( E$ k- | }3 z3 T

8 W4 t- z: q& u# q9 J* J3 L3 m

means that for every neighborhood N₁ ( A ) there is some neighborhood N₂ ( p) such that

f ( x ) ∈ N₁ ( A ) whenever x ∈ N₂ ( p ) and x ≠ p (*) 2 \6 J6 f B' Y; p3 U4 s

4 Y5 n8 K: A* N% P6 y1 C

! m4 o4 |1 E' P8 [. d# M& j% C

The first thing to note about this definition is that it involves two neighborhoods, N₁ ( A) and

N₂ ( p) . The neighborhood N₁ ( A) is specified first; it tells us how close we wish f ( x ) to be to the limit A. The second neighborhood, N₂ ( p ), tells us how close x should be to p so that f ( x ) will be within the first neighborhood N₁ ( A). The essential part of the definition is that, for every N₁ ( A), no matter how small, there is some neighborhood N₂ (p) to satisfy (*). In general, the neighborhood N₂ ( p) will depend on the choice of N₁ ( A). A neighborhood N₂ ( p ) that works for one particular N₁ ( A) will also work, of course, for every larger N₁ ( A), but it may not be suitable for any smaller N₁ ( A).

The definition of limit can also be formulated in terms of the radii of the neighborhoods

N₁ ( A) and N₂ ( p ). It is customary to denote the radius of N₁ ( A) byεand the radius of N₂ ( p) by δ.The statement f ( x ) ∈ N₁ ( A ) is equivalent to the inequality ∣f ( x ) – A∣<ε,and the statement x ∈ N₁ ( A) ,x ≠ p ,is equivalent to the inequalities 0 <∣ x-p∣<δ. Therefore, the definition of limit can also be expressed as follows:

The symbol f ( x ) = A means that for everyε> 0, there is aδ> 0 such that

∣f ( x ) – A∣<ε whenever 0 <∣x – p∣<δ

“One-sided” limits may be defined in a similar way. For example, if f ( x ) →A as x→ p through values greater than p, we say that A is right-hand limit of f at p, and we indicate this by writing

f ( x ) = A 3 v7 k6 |+ Y& S( o% }5 U

$ N$ @/ a3 s% F. E3 d; B+ @

1 i4 n( ?: j: N6 J4 e7 t

In neighborhood terminology this means that for every neighborhood N₁ ( A) ,there is some neighborhood N₂( p) such that

f ( x ) ∈ N₁ ( A) whenever x ∈ N₁ ( A) and x > p ; T9 W0 q: I0 L

2 t- `+ N/ t0 {1 [% s( L

/ A' E1 E% B8 X W1 Q

Left-hand limits, denoted by writing x→ p^-, are similarly defined by restricting x to values less than p.

If f has a limit A at p, then it also has a right-hand limit and a left-hand limit at p, both of these being equal to A. But a function can have a right-hand limit at p different from the left-hand limit.

The definition of continuity of a function.

2 B+ x$ _" K1 p, [# u l+ O

! k1 M- ~8 g" ~( r

9 W6 d( d9 |* d, ~3 w3 Y5 F! q

6 D8 c+ i& v6 `

作者: hehe123 时间: 2004-11-27 12:30
In the definition of limit we made no assertion about the behaviour of f at the point p itself. Moreover, even if f is defined at p, its value there need not be equal to the limit A. However, if it happens that f is defined at p and if it also happens that f ( p ) = A, then we say the function f is continuous at p. In other words, we have the following definition.

Definition of continuity of a function at a point.

A function f is said to be continuous at a point p if

( a ) f is defined at p, and ( b ) f ( x ) = f ( p ) ) E3 ~ M! X! l/ g

This definition can also be formulated in term of neighborhoods. A function f is continuous at p if for every neighborhood N₁ ( f(p)) there is a neighborhood N₂ (p) such that

f ( x ) ∈ N₁ ( f (p)) whenever x ∈ N₂ ( p).

In theε-δterminology , where we specify the radii of the neighborhoods, the definition of continuity can be restated ad follows:

Function f is continuous at p if for every ε> 0 ,there is aδ> 0 such that

∣f ( x ) – f ( p )∣< ε whenever ∣x – p∣< δ

In the rest of this lesson we shall list certain special properties of continuous functions that are used quite frequently. Most of these properties appear obvious when interpreted geometrically ; consequently many people are inclined to accept them ad self-evident. However, it is important to realize that these statements are no more self-evident than the definition of continuity itself, and therefore they require proof if they are to be used with any degree of generality. The proofs of most of these properties make use of the least-upper bound axiom for the real number system.

THEOREM 1. (Bolzano’s theorem) Let f be continuous at each point of a closed interval [a, b] and assume that f ( a ) an f ( b ) have opposite signs. Then there is at least one c in the open interval (a ,b) such that f ( c ) = 0.

THEOREM 2. Sign-preserving property of continuous functions. Let f be continuious at c and suppose that f ( c ) ≠ 0. Then there is an interval (c-δ,c +δ) about c in which f has the same sign as f ( c ).

THEOREM 3. Let f be continuous at each point of a closed interval [a, b]. Choose two arbitrary points x₁ <x₂ in [a, b] such that f ( x₁) ≠ f ( x₂) . Then f takes every value between f ( x₁ ) and f (x₂ ) somewhere in the interval ( x_1, x₂).

THEOREM 4. Boundedness theorem for continuous functions. Let f be continuous on a closed interval [a, b]. Then f is bounded on [a, b]. That is , there is a number M > 0, such that∣f ( x )∣≤ M for all x in [a, b].

THEOREM 5. (extreme value theorem) Assume f is continuous on a closed interval [a, b]. Then there exist points c and d in [a, b] such that f ( c ) = sup f and f ( d ) = inf f .

Note. This theorem shows that if f is continuous on [a, b], then sup f is its absolute maximum, and inf f is its absolute minimum

作者: hehe123 时间: 2004-11-27 12:31

Vocabulary 4 _- B G3 U/ {. P9 `" W/ T

continuity 连续性 assume 假定,取

continuous 连续的 specify 指定, 详细说明

continuous function 连续函数 statement 陈述,语句

intuitive 直观的 right-hand limit 右极限

corresponding 对应的 left-hand limit 左极限

correspondence 对应 restrict 限制于

graph 图形 assertion 断定

approach 趋近,探索,入门 consequently 因而,所以

tend to 趋向 prove 证明

regardless 不管,不顾 proof 证明

discontinuous 不连续的 bound 限界

jump discontinuity 限跳跃不连续 least upper bound 上确界

mathematician 科学家 greatest lower bound 下确界

formulate 用公式表示,阐述 boundedness 有界性

limit 极限 maximum 最大值

Interval 区间 minimum 最小值

open interval 开区间 extreme value 极值

equation 方程 extremum 极值

neighborhood 邻域 increasing function 增函数

midpoint 中点 decreasing function 减函数

symmetric 对称的 strict 严格的

radius 半径(单数) uniformly continuous 一致连续

radii 半径(复数) monotonic 单调的

inequality 不等式 monotonic function 单调函数

equivalent 等价的

作者: hehe123 时间: 2004-11-27 12:31

Notes ( D1 ~3 Z7 L0 T, z, A. N

1. It wad not until late in the 18th century that discontinuous functions began appearing in connection with various kinds of physical problems.

意思是:直到十八世纪末,不连续函数才开始出现于与物理学有关的各类问题中.

这里It was not until …that译为“直到……才”

2. The symbol f ( x ) = A means that for every ε> 0 ,there is a δ> 0, such that

|f( x ) - A|<ε whenever 0 <| x – p |<δ

注意此种句型.凡涉及极限的其它定义,如本课中定义函数在点P连续及往后出现的关于收敛的定义等,都有完全类似的句型,参看附录IV.

有时句中there is可换为there exists; such that可换为satisfying; whenever换成if或for.

3. Let…and assume (suppose)…Then…

这一句型是定理叙述的一种最常见的形式;参看附录IV.一般而语文课 Let假设条件的大前提,assume (suppose)是小前提(即进一步的假设条件),而if是对具体而关键的条件的使用语.

4. Approach在这里是“趋于”,“趋近”的意思,是及物动词.如:

f ( x ) approaches A as x approaches p. Approach有时可代以tend to. 如f ( x ) tends to A as x tends to p.值得留意的是approach后不加to而tend之后应加to.

5. as close to A as we please = arbitrarily close to A..

作者: hehe123 时间: 2004-11-27 12:34

Exercise 1 t: v# A1 \# O4 {5 W! n/ t+ ?

I. Fill in each blank with a suitable word to be chosen from the words given below:

independent domain correspondence

associates variable range

(a) Let y = f ( x ) be a function defined on [a, b]. Then

(i) x is called the ____________variable.

(ii) y is called the dependent ___________.

(iii) The interval [a, b] is called the ___________ of the function.

(b) In set terminology, the definition of a function may be given as follows:

Given two sets X and Y, a function f : X → Y is a __________which ___________with each element of X one and only one element of Y.

II. a) Which function, the exponential function or the logarithmic function, has the property that it satisfies the functional equation

f ( xy ) = f ( x ) + f ( v )

b) Give the functional equation which will be satisfied by the function which you do not choose in (a).

III. Let f be a real-valued function defined on a set S of real numbers. Then we have the following two definitions:

i) f is said to be increasing on the set S if f ( x ) < f ( y ) for every pair of points x and y with x < y.

ii) f is said to have and absolute maximum on the set S if there is a point c in S such that f ( x ) < f ( c ) for all x∈ S.

Now define

a) a strictly increasing function;

b) a monotonic function;

c) the relative (or local ) minimum of f .

IV. Translate theorems 1-3 into Chinese.

V. Translate the following definition into English:

定义:设E 是定义在实数集 E 上的函数,那么, 当且仅当对应于每一ε>0(ε不依赖于E上的点)存在一个正数δ使得当 p 和 q 属于E且|p –q| <δ时有|f ( p ) – f ( q )|<ε,则称f在E上一致连续.

作者: hehe123 时间: 2004-11-27 12:34
标题: 数学专业英语[4]－Differential Calculus

Historical Introduction

Newton and Leibniz,quite independently of one another,were largely responsible for developing the ideas of integral calculus to the point where hitherto insurmountable problems could be solved by more or less routine methods.The successful accomplishments of these men were primarily due to the fact that they were able to fuse together the integral calculus with the second main branch of calculus,differential calculus.

The central idea of differential calculus is the notion of derivative.Like the integral,the derivative originated from a problem in geometry—the problem of finding the tangent line at a point of a curve.Unlile the integral,however,the derivative evolved very late in the history of mathematics.The concept was not formulated until early in the 17^th century when the French mathematician Pierre de Fermat,attempted to determine the maxima and minima of certain special functions.

Fermat’s idea,basically very simple,can be understood if we refer to a curve and assume that at each of its points this curve has a definite direction that can be described by a tangent line.Fermat noticed that at certain points where the curve has a maximum or minimum,the tangent line must be horizontal.Thus the problem of locating such extreme values is seen to depend on the solution of another problem,that of locating the horizontal tangents.

This raises the more general question of determining the direction of the tangent line at an arbitrary point of the curve.It was the attempt to solve this general problem that led Fermat to discover some of the rudimentary ideas underlying the notion of derivative.

At first sight there seems to be no connection whatever between the problem of finding the area of a region lying under a curve and the problem of finding the tangent line at a point of a curve.The first person to realize that these two seemingly remote ideas are,in fact, rather intimately related appears to have been Newton’s teacher,Isaac Barrow(1630-1677).However,Newton and Leibniz were the first to understand the real importance of this relation and they exploited it to the fullest,thus inaugurating an unprecedented era in the development of mathematics.

Although the derivative was originally formulated to study the problem of tangents,it was soon found that it also provides a way to calculate velocity and,more generally,the rate of change of a function.In the next section we shall consider a special problem involving the calculation of a velocity.The solution of this problem contains all the essential fcatures of the derivative concept and may help to motivate the general definition of derivative which is given below.

A Problem Involving Velocity

Suppose a projectile is fired straight up from the ground with initial velocity of 144 feet persecond.Neglect friction,and assume the projectile is influenced only by gravity so that it moves up and back along a straight line.Let f(t) denote the height in feet that the projectile attains t seconds after firing.If the force of gravity were not acting on it,the projectile would continue to move upward with a constant velocity,traveling a distance of 144 feet every second,and at time t we woule have f(t)=144 t.In actual practice,gravity causes the projectile to slow down until its velocity decreases to zero and then it drops back to earth.Physical experiments suggest that as the projectile is aloft,its height f(t) is given by the formula

(1) f(t)=144t –16 t²

The term –16t² is due to the influence of gravity.Note that f(t)=0 when t=0 and when t=9.This means that the projectile returns to earth after 9 seconds and it is to be understood that formula (1) is valid only for 0<t<9.

The problem we wish to consider is this:To determine the velocity of the projectile at each instant of its motion.Before we can understand this problem,we must decide on what is meant by the velocity at each instant.To do this,we introduce first the notion of average velocity during a time interval,say from time t to time t+h.This is defined to be the quotient.

Change in distance during time interval =f(t+h)-f(t)/h

Length of time interval

This quotient,called a difference quotient,is a number which may be calculated whenever both t and t+h are in the interval[0,9].The number h may be positive or negative,but not zero.We shall keep t fixed and see what happens to the difference quotient as we take values of h with smaller and smaller absolute value.

The limit process by which v(t) is obtained from the difference quotient is written symbolically as follows:

V(t)=lim_(h_→0) [f(t+h)-f(t)]/h

The equation is used to define velocity not only for this particular example but,more generally,for any particle moving along a straight line,provided the position function f is such that the differerce quotient tends to a definite limit as h approaches zero.

The example describe in the foregoing section points the way to the introduction of the concept of derivative.We begin with a function f defined at least on some open interval(a,b) on the x axis.Then we choose a fixed point in this interval and introduce the difference quotient

[f(x+h)-f(x)]/h

where the number h,which may be positive or negative(but not zero),is such that x+h also lies in(a,b).The numerator of this quotient measures the change in the function when x changes from x to x+h.The quotient itself is referred to as the average rate of change of f in the interval joining x to x+h.

Now we let h approach zero and see what happens to this quotient.If the quotient.If the quotient approaches some definite values as a limit(which implies that the limit is the same whether h approaches zero through positive values or through negative values),then this limit is called the derivative of f at x and is denoted by the symbol f’(x) (read as “f prime of x”).Thus the formal definition of f’(x) may be stated as follows:

Definition of derivative.The derivative f’(x)is defined by the equation

f’(x)=lim_(h_→o)[f(x+h)-f(x)]/h

provided the limit exists.The number f’(x) is also called the rate of change of f at x.

In general,the limit process which produces f’(x) from f(x) gives a way of obtaining a new function f’ from a given function f.This process is called differentiation,and f’ is called the first derivative of f.If f’,in turn,is defined on an interval,we can try to compute its first derivative,denoted by f’’,and is called the second derivative of f.Similarly,the nth derivative of f denoted by f^(n),is defined to be the first derivative of f^(n-1).We make the convention that f^(0)=f,that is,the zeroth derivative is the function itself.

作者: hehe123 时间: 2004-11-27 12:35

Vocabulary

differential calculus微积分 differentiable可微的

intergral calculus 积分学 differentiate 求微分

hither to 迄今 integration 积分法

insurmountable 不能超越 integral 积分

routine 惯常的 integrable 可积的

fuse 融合 integrate 求积分

originate 起源于 sign-preserving保号

evolve 发展，引出 axis 轴（单数）

tangent line 切线 axes 轴（复数）

direction 方向 contradict 矛盾

horizontal 水平的 contradiction 矛盾

vertical 垂直的 contrary 相反的

rudimentary 初步的，未成熟的 composite function 合成函数，复合函数

area 面积 composition 复合函数

intimately 紧密地 interior 内部

exploit 开拓，开发 interior point 内点

inaugurate 开始 imply 推出，蕴含

projectile 弹丸 aloft 高入云霄

friction摩擦 initial 初始的

gravity 引力 instant 瞬时

rate of change 变化率 integration by parts分部积分

attain 达到 definite integral 定积分

defferential 微分 indefinite integral 不定积分

differentiation 微分法 average 平均

作者: hehe123 时间: 2004-11-27 12:35

Notes 4 g! l7 k" Y, o% G2 X

1. Newton and Leibniz,quite independently of one another,were largely responsible for developing…by more or less routine methods.

意思是：在很大程度上是牛顿和莱伯尼，他们相互独立地把积分学的思想发展到这样一种程度，使得迄今一些难于超越的问题可以或多或少地用通常的方法加以解决。

这里responsible for的基本意义是：“对…负责”，但也可作“应归功于”解，这里应理解为“归功于”。

2．The example described in the foregoing section points the way to the introduction of the concept of derivative.

意思是：前面一节所描述的例子指出了引进导数概念的方法。

这里described是过去分词，foregoing是现在分词，两者都用作定语，切不可误认described为过去式谓语。类似句子如：

We begin with a function f defined on some interval(a,b);

3. The quotient itself is referred to as the average of change of f in the interval joining x to x+h.

意思是：商本身是指区间x到x+h上f的平均变化率。这里be referred to as意思是：“把…

认为是“

4．We make the convention that f⁰=f

意思是：我们约定（按惯例）f⁰=f

作者: hehe123 时间: 2004-11-27 12:36

Exercise ) f3 |$ r1 G* p1 F3 e" |3 F

I.1.Fill in the missing words in column B such that the word in column B corresponds to the word in column A in the same sense as “integration”corresponds to “differentiation”

A B

Differentiation integration

Differential

Differentiate

Differentiable

1. Then choose the correct word from either column A or column B and insert it in each of the blanks.

(i)The process of finding the derivative of a function is called( )

(ii) If f(x) has a derivative at the point x₀then f(x) is said to be ( )at x_0.

(iii) ∫f(x)dx is called the indefinite( ) of f(x).

II Translate the following two examples into Chinese(pay attention to the phrases used):

Example 1 Find the derivative of

f(x)=(3x+1)⁴/(x²+2)³

solution:Taking the logarithms of both sides,we have

ln f(x)=4ln(3x+1)-3ln(x²+2)

Differentiating both sides of the above equation,we obtain

f’(x)/f(x)=12/(3x+1)-6/(x²+2)

(1) and (2)together yield

f’(x)=[(3x+1)⁴/(x2+2)³][12/(3x+1)-6/(x²+2)]

Example 2 Integrate ∫ x² cosxdx

Solutionet u=x²,v=sin x;

Then du=2xdx,dv=cos x dx

So we have I=∫x² cos x dx =∫u dv

By applying integration by parts,we have

I=∫udv=uv-∫vdu=x² sinx-2∫x sinx dx (1)

Applying integration by parts once again to the indefinite integral∫xsinxdx,we get

∫x sinx dx = -x cos x +sin x +c (2)

Substituting (2)into (1)yields

∫ x² cos x dx=x² sin x+2x cos x-2 sin x +c

where c is an arbitrary constant.

III. Translate the following example into English:

求y=ln(2x²-4)的导数

【解】令y=ln u,u=(2x2-4),则dy/du=1/u. du/dx=4x (1)

据复合函数求导数的公式，我们有

dy/du=dy/du-du/dx (2)

把（1）代入（2）式得dy/dx=(1/u)4x

把u换为2x²-4,最后得dy/dx=4x/(2x²-4)

IV. Theorem Let f be defined on an open interval I,and assume that f has a relative maximum or a relative minimum at an interior point c of I.If the derivative f’(c )exists,then f’(c )=0

The proof of this theorem is given in Chinese as follows.Turn it into English:

证明：

（1）在I上定义函数Q（x）

Q(x)=[f(x)-f(c )]/(x-c) x≠c

f’(c ) x=c

(2) 因为f’(c)存在，故当x趋向c时，Q（x）趋向Q（c），也即Q（x）在点x＝c连续

（2）我们将证明：若f’（c）＝Q（c）≠0，则导致矛盾；

（3）设Q（c）>0，根据连续函数的保号性质，存在c点的一个领域，在此领域里，Q（x）是正的；

（4）因此在此领域内，对所有x≠c，Q（x）的分子和分母同号；

（5）即是说，当x>c时，f（x）>f(c ),而当x<c时，f（x）<f（c）这与f在x＝c处有一极值相矛盾；

（6）因此Q（c）>0不可能，同理可证Q（c）<0也不真。

作者: hehe123 时间: 2004-11-27 12:37
标题: 数学专业英语[5]－First Order Differential Equations

A differential equation is an equation between specified derivatives of a function, its

valves,and known quantities.Many laws of physics are most simply and naturally formu-

lated as differential equations (or DE’s, as we shall write for short).For this reason,DE’s

have been studies by the greatest mathematicians and mathematical physicists since the

time of Newton..

Ordinary differential equations are DE’s whose unknowns are functions of a single va-

riable;they arise most commonly in the study of dynamic systems and electric networks.

They are much easier to treat than partial differential equations,whose unknown functions

depend on two or more independent variables.

Ordinary DE’s are classified according to their order. The order of a DE is defined as

the largest positive integer, n, for which an n-th derivative occurs in the equation. This

chapter will be restricted to real first order DE’s of the form

Φ(x, y, y′)=0 (1)

Given the function Φof three real variables, the problem is to determine all real functions y=f(x) which satisfy the DE, that is ,all solutions of(1)in the following sense.

DEFINITION A solution of (1)is a differentiable function f(x) such that

Φ(x. f(x),f′(x))=0 for all x in the interval where f(x) is defined.

EXAMPLE 1. In the first-other DE

x+yy′=0 (2)

the function Φ is a polynomial function Φ(x, y, z)=x+ yz of three variables in-

volved. The solutions of (2) can be found by considering the identity

d(x²+y²)/d x=2(x+yyˊ).From this identity,one sees that x²+y² is a con-

stant if y=f(x) is any solution of (2).

The equation x²+y²=c defines y implicitly as a two-valued function of x,

for any positive constant c.Solving for y,we get two solutions,the(single-valued)

functions y=±(c-x²)^0.5 ,for each positive constant c.The graphs of these so-

lutions,the so-called solution curves,form two families of scmicircles,which fill the upper half-plane y>0 and the lower half-plane y>0,respectively.

On the x-axis,where y=0,the DE(2) implies that x=0.Hence the DE has no solutions

which cross the x-axis,except possibly at the origin.This fact is easily overlooked,

because the solution curves appear to cross the x-axis;hence yˊdoes not exist,and the DE (2) is not satisfied there.

The preceding difficulty also arises if one tries to solve the DE(2)for yˊ. Dividing through by y,one gets yˊ=-x/y,an equation which cannot be satisfied if y=0.The preceding difficulty is thus avoided if one restricts attention to regions where the DE(1) is normal,in the following sense.

DEFINITION. A normal first-order DE is one of the form

yˊ=F(x,y) (3)

In the normal form yˊ=-x/y of the DE (2),the function F(x,y) is continuous in the upper half-plane y>0 and in the lower half-plane where y<0;it is undefined on the x-axis.

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Fundamental Theorem of the Calculus. , E/ h7 @3 K& m. N

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The most familiar class of differential equations consists of the first-order DE’s of the form

yˊ=g(x) (4)

Such DE’s are normal and their solutions are descried by the fundamental thorem of the calculus,which reads as follows.

FUNDAMENTAL THEOREM OF THE CALCULUS. Let the function g(x)in DE(4) be continuous in the interval a<x<b.Given a number c,there is one and only one solution f(x) of the DE(4) in the interval such that f(a)=c. This solution is given by the definite integral

f(x)=c+∫_a^xg(t)dt , c=f(a) (5)

This basic result serves as a model of rigorous formulation in several respects. First,it specifies the region under consideration,as a vertical strip a<x<b in the xy-plane.Second,it describes in precise terms the class of functions g(x) considered.And third, it asserts the existence and uniqueness of a solution,given the “initial condition”f(a)=c.

We recall that the definite integral

∫_a^xg(t)dt=lim(maxΔt_k->0)Σg(t_k)Δt_k, Δt_k=t_k-t_k-1 (5ˊ)

is defined for each fixed x as a limit of Ricmann sums; it is not necessary to find a formal expression for the indefinite integral ∫ g(x) dx to give meaning to the definite integral ∫_a^xg(t)dt,provided only that g(t) is continuous.Such functions as the error function crf x =(2/(π)^0.5)∫₀^xe^-t^²dt and the sine integral function SI(x)=∫_x^∞[(sin t )/t]dt are indeed commonly defined as definite integrals.

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Solutions and Integrals + S" X6 H1 p$ a2 I* \. r

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According to the definition given above a solution of a DE is always a function. For example, the solutions of the DE x+yyˊ=0 in Example I are the functions y=± (c-x²)^0.5,whose graphs are semicircles of arbitrary diameter,centered at the origin.The graph of the solution curves are ,however,more easily described by the equation x²+y²=c,describing a family of circles centered at the origin.In what sense can such a family of curves be considered as a solution of the DE ?To answer this question,we require a new notion.

DEFINITION. An integral of DE(1)is a function of two variables,u(x,y),which assumes a constant value whenever the variable y is replaced by a solution y=f(x) of the DE.

In the above example, the function u(x,y)=x²+y² is an integral of the DE x+yyˊ =0,because,upon replacing the variable y by any function ±( c-x²)^0.5,we obtain u(x,y)=c.

The second-order DE

d²x/dt²=-x (2ˊ)

becomes a first-order DE equivalent to (2) after setting dx/dx=y:

y ( dy/dx )=-x (2)

As we have seen, the curves u(x,y)=x²+y²=c are integrals of this DE.When the DE (2ˊ)

is interpreted as equation of motion under Newton’s second law,the integrals

c=x²+y² represent curves of constant energy c.This illustrates an important principle:an integral of a DE representing some kind of motion is a quantity that remains unchanged through the motion.

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作者: hehe123 时间: 2004-11-27 12:37

Vocabulary 5 U% H4 Q$ c# ]' J! O

differential equation 微分方程 error function 误差函数

ordinary differential equation 常微分方程 sine integral function 正弦积分函数

order 阶,序 diameter 直径

derivative 导数 curve 曲线

known quantities 已知量 replace 替代

unknown 未知量 substitute 代入

single variable 单变量 strip 带形

dynamic system 动力系统 exact differential 恰当微分

electric network 电子网络 line integral 线积分

partial differential equation 偏微分方程 path of integral 积分路径

classify 分类 endpoints 端点

polynomial 多项式 general solution 通解

several variables 多变量 parameter 参数

family 族 rigorous 严格的

semicircle 半圆 existence 存在性

half-plane 半平面 initial condition 初始条件

region 区域 uniqueness 唯一性

normal 正规,正常 Riemann sum 犁曼加

identity 恒等(式)

作者: hehe123 时间: 2004-11-27 12:38

Notes % q. U2 ?4 \# [; K8 e; C. q; \

1. The order of a DE is defined as the largest positive integral n,for which an nth derivative occurs in the question.

这是另一种定义句型,请参看附录IV.此外要注意nth derivative 之前用an 不用a .

2. This chapter will be restricted to real first order differential equations of the form

Φ(x,y,yˊ)=0

意思是;文章限于讨论形如 Φ(x,y,yˊ)=0的实一阶微分方程.

有时可以用of the type代替 of the form 的用法.

The equation can be rewritten in the form yˊ=F(x,y).

3. Dividing through by y,one gets yˊ=-x/y,…

划线短语意思是:全式除以y

4. As we have seen, the curves u(x,y)=x²+y²=c are integrals of this DE

这里x²+y²=c 因c是参数,故此方程代表一族曲线,由此”曲线”这一词要用复数curves.

5. Their solutions are described by the fundamental theorem of the calculus,which reads as follows.

意思是:它们的解由微积分基本定理所描述,(基本定理)可写出如下.

句中reads as follows 就是”写成(读成)下面的样子”的意思.注意follows一词中的”s”不能省略.

作者: hehe123 时间: 2004-11-27 12:38

Exercise 0 b. X- H ]7 Q6 M. E. y/ L( _

Ⅰ.Translate the following passages into Chinese:

1.A differential M(x,y) dx +N(x,y) dy ,where M, N are real functions of two variables x and y, is called exact in a domain D when the line integral ∫_cM(x,y) dx +N(x,y) dy is the same for all paths of integration c in D, which have the same endpoints.

Mdx+Ndy is exact if and only if there exists a continuously differentiable function u(x,y) such that M= u/ x, N=u/ y.

2. For any normal first order DE yˊ=F(x,y) and any initial x₀, the initial valve problem consists of finding the solution or solutions of the DE ,for x>x₀which assumes a given initial valve f(x₀)=c.

3. To show that the initial valve problem is well-set requires proving theorems of existence (there is a solution), uniqueness (there is only one solution) and continuity (the solution depends continuously on the initial value).

Ⅱ. Translate the following sentences into English:

1) 因为y=ч(x) 是微分方程dy/ dx=f(x,y)的解,故有

dч(x)/dx=f (x,ч(x))

2) 两边从x₀到x取定积分得

ч(x)-ч(x₀)=∫_x0^xf(x,ч(x)) dx x₀<x<x₀+h

3) 把y₀=ч(x₀)代入上式, 即有

ч(x)=y₀+∫_x0^xf(x,ч(x)) dx x₀<x<x₀+h

4) 因此 y=ч(x) 是积分方程

y=y₀+∫_x0^x f (x,y) dx

定义于x₀<x<x₀+h 的连续解.

Ⅲ. Translate the following sentences into English:

1) 现在讨论型如 y=f (x,yˊ) 的微分方程的解,这里假设函数 f (x, dy/dx) 有连续的偏导数.

2) 引入参数dy/dx=p, 则已给方程变为 y=f (x,p).

3) 在 y=f (x,p) x p=dy/dx p= f/ x+f/ p dp/dx

4) 这是一个关于x和p的一阶微分方程,它的解法我们已经知道.

5) 若(A)的通解的形式为p=ч(x,c) ,则原方程的通解为

y=f (x,ч(x,c)).

6) 若(A) 有型如x=ψ(x,c)的通解,则原方程有参数形式的通解

x=ψ(p,c)

y=f(ψ(p,c)p)

其中p是参数,c是任意常数.

作者: hehe123 时间: 2004-11-27 12:45
标题: 数学专业英语[6]－Sequences and Series

Series are a natural continuation of our study of functions. In the previous chapter we found how

to approximate our elementary functions by polynomials, with a certain error term. Conversely, one can define arbitrary functions by giving a series for them. We shall see how in the sections below.

In practice, very few tests are used to determine convergence of series. Essentially, the comparision test is the most frequent. Furthermore, the most important series are those which converge absolutely. Thus we shall put greater emphasis on these.

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Convergent Series

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Suppose that we are given a sequcnce of numbers

a₁,a₂,a₃…

i.e. we are given a number a_n, for each integer ｎ＞1.We form the sums

S_n=a₁+a₂+…+a_{n
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It would be meaningless to form an infinite sum

a₁+a₂+a₃+…

because we do not know how to add infinitely many numbers. However, if our sums S_n approach a limit as n becomes large, then we say that the sum of our sequence converges, and we now define its sum to be that limit.

The symbols

∑_a=1^∞a_n^{- O5 O# w) U+ F3 J- I# Z0 C( Y& s}

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will be called a series. We shall say that the series converges if the sums approach a limit as n becomes large. Otherwise, we say that it does not converge, or diverges. If the seriers converges, we say that the value of the series is

∑_a=1^∞=lim_a_→∞S_n=lim_a_→∞(a₁+a₂+…+a_n)

In view of the fact that the limit of a sum is the sum of the limits, and other standard properties of limits, we get:

THEOREM 1. Let{ a_n }and { b_n }(n=1,2,…)

be two sequences and assume that the series

∑_a=1^∞a_n∑_a=1^∞b_n^{1 J! V3 D7 X( t+ N" n3 [+ j& u* |}

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converge. Then ∑_a=1^∞(a_n+ b_n) also converges, and is equal to the sum of the two series. If c is a number, then

∑ _a=1^∞c a_n =c∑_a=1 ^∞a_{n 8 _0 t3 G$ a7 V}

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Finally, if s_n=a₁+a₂+…+a_n and t_n=b₁+b₂+…+b_n then

∑_a=1^∞a_n∑_a=1^∞b_n=lim_a_→∞s_nt_n

In particular, series can be added term by term. Of course , they cannot be multiplied term by term.

We also observe that a similar theorem holds for the difference of two series.

If a series ∑a_n converges, then the numbers a_n must approach 0 as n becomes large. However, there are examples of sequences {an} for which the series does not converge, and yet lim_a_→∞a_n=0

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Series with Positive Terms

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Throughout this section, we shall assume that our numbers a_n are ＞ 0. Then the partial sums

S_n=a₁+a₂+…+a_n

are increasing, i.e.

s₁＜s₂＜s₃＜…＜s_n＜s_n+1＜…

If they are approach a limit at all, they cannot become arbitrarily large. Thus in that case there is a number B such that

S_n< B

for all n. The collection of numbers {s_n} has therefore a least upper bound ,i.e. there is a smallest number S such that

s_n<S

for all n. In that case , the partial sums s_n approach S as a limit. In other words, given any positive number ε>0, we have

S –ε< s_n< S

for all n .sufficiently large. This simply expresses the fact that S is the least of all upper bounds for our collection of numbers s_n. We express this as a theorem.

THEOREM 2. Let{a_n}(n=1,2,…)be a sequence of numbers>0 and let

S_n=a₁+a₂+…+a_n

If the sequence of numbers {s_n} is bounded, then it approaches a limit S , which is its least upper bound.

Theorem 3 gives us a very useful criterion to determine when a series with positive terms converges:

THEOREM 3. Let∑_a=1^∞a_n and∑_a=1^∞ b_n be two series , with a_n>0 for all n and b_n>0 for all n. Assume that there is a number c such that

a_n< c b_n

for all n, and that∑_a=1^∞b_n converges. Then ∑_a=1^∞ a_n converges, and

∑_a=1^∞a_n≤ c∑_a=1^∞b_n

PROOF. We have

a₁+…+a_n≤cb₁+…+cb_n

=c(b₁+…+b_n)≤ c∑_a=1^∞b_n

This means that c∑_a=1^∞b_n is a bound for the partial sums a₁+…+a_n.The least upper bound of these sums is therefore ≤ c∑_a=1^∞b_n, thereby proving our theorem.

Differentiation and Intergration of Power Series.

If we have a polynomial

a₀+a₁x+…+a_nxⁿ

with numbers a₀,a₁,…,a_n as coefficients, then we know how to find its derivative. It is a₁+2a₂x+…+na_nxⁿ^–1. We would like to say that the derivative of a series can be taken in the same way, and that the derivative converges whenever the series does.

THEOREM 4. Let r be a number >0 and let ∑a_nxⁿ be a series which converges absolutely for ∣x∣<r. Then the series ∑na_nx^n-1 also converges absolutely for∣x∣<r.

A similar result holds for integration, but trivially. Indeed, if we have a series ∑_a=1^∞a_nxⁿwhich converges absolutely for ∣x∣<r, then the series

∑_a=1^∞a_n/n+1 xⁿ⁺¹=x∑_a=1^∞a_nxⁿ∕n+1

has terms whose absolute value is smaller than in the original series.

The preceding result can be expressed by saying that an absolutely convergent series can be integrated and differentiated term by term and and still yields an absolutely convergent power series.

It is natural to expect that if

f (x)=∑_a=1^∞a_nxⁿ,

then f is differentiable and its derivative is given by differentiating the series term by term. The next theorem proves this.

THEOREM 5. Let

f (x)=∑_a=1^∞ a_nxⁿ

be a power series, which converges absolutely for∣x∣<r. Then f is differentiable for ∣x∣<r, and

f′(x)=∑_a=1^∞na_nx^n-1.

THEOREM 6. Let f (x)=∑_a=1^∞a_nxⁿ be a power series, which converges absolutely for ∣x∣<r. Then the relation

∫f (x)d x=∑_a=1^∞a_nxⁿ⁺¹∕n+1

is valid in the interval ∣x∣<r.

We omit the proofs of theorems 4,5 and 6.

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作者: hehe123 时间: 2004-11-27 12:46

Vocabulary ) D4 i$ ]8 y! u V. e3 c

sequence 序列 positive term 正项

series 级数 alternate term 交错项

approximate 逼近,近似 partial sum 部分和

elementary functions 初等函数 criterion 判别准则(单数)

section 章节 criteria 判别准则(多数)

convergence 收敛(名词) power series 幂级数

convergent 收敛(形容词) coefficient 系数

absolute convergence 绝对收敛 Cauchy sequence 哥西序列

diverge 发散 radius of convergence 收敛半径

term by term 逐项 M-test M—判别法

作者: hehe123 时间: 2004-11-27 12:46

Notes $ L! g. N5 k4 p' P% `1 z0 Z* G

1. series一词的单数和复数形式都是同一个字.例如:

One can define arbitrary functions by giving a series for them(单数)

The most important series are those which converge absolutely(复数)

2. In view of the fact that the limit of a sum of the limits, and other standard properties of limits, we get:

Theorem 1…

这是叙述定理的一种方式: 即先将事实说明在前面,再引出定理. 此句用in view of the fact that 说明事实,再用we get 引出定理.

3. We express this as a theorem.

这是当需要证明的事实已再前面作了说明或加以证明后,欲吧已证明的事实总结成定理时,常用倒的一个句子,类似的句子还有(参看附录Ⅲ):

We summarize this as the following theorem; Thus we come to the following theorem等等.

4. The least upper bound of these sums is therefore ≤c∑_a=1^∞b_n, thereby proving our theorem.

最一般的定理证明格式是”给出定理…定理证明…定理证毕”,即thereby proving our theorem;或we have thus proves the theorem或This completes the proof等等作结尾(参看附录Ⅲ).

5. 本课文使用较多插入语.数学上常见的插入语有:conversely; in practice; essentially; in particular; indeed; in other words; in short; generally speaking 等等.插入语通常与句中其它成份没有语法上的关系,一般用逗号与句子隔开,用来表示说话者对句子所表达的意思的态度.插入语可以是一个词,一个短语或者一个句子.

作者: hehe123 时间: 2004-11-27 12:46
Exercise

Ⅰ. Translate the following exercises into Chinese:

1. In exercise 1 through 4,a sequence f (n) is defined by the formula given. In each case, (ⅰ)

Determine whether the sequence (the formulae are omitted).

2. Assume f is a non–negative function defined for all x>1. Use the method

suggested by the proof of the integral test to show that

∑_k=1^n-1f(k)≤∫₁ⁿf(x)d x ≤∑_k=2ⁿf(k)

Take f(x)=log x and deduce the inequalities

c•nⁿ•c^-n< n！<c•nⁿ⁺¹•c^-n

Ⅱ. The proof of theorem 4 is given in English as follows(Read the proof through and try to learn how a theorem is proved, then translate this proof into Chinese ):

Proof of theorem 4 Since we are interested in the absolute convergence. We may assume that a_n>0 for all n. Let 0<x<r, and let c be a number such that x<c<r. Recall that lim_a_→∞n^1/n=1.

We may write n a_nxⁿ=a_n(n^1/nx)ⁿ. Then for all n sufficiently large, we conclude that n^1/nx<c. This is because n1/n comes arbitrarily close to x and x<c. Hence for all n sufficiently large, we have na_nxⁿ<a_ncⁿ. We can then compare the series ∑na_nxⁿwith∑a_ncⁿ to conclude that∑na_nxⁿ converges. Since∑na_nx^n-1=1/x∑na_nxⁿ, we have proved theorem 4.

Ⅲ. Recall from what you have learned in Calculus about (ⅰ) Cauchy sequence and (ⅱ) the radius of convergence of a power series.

Now give the definitions of these two terms respectively.

Ⅳ. Translate the following sentences into Chinese:

1. 一旦我们能证明,幂级数∑a_nzⁿ 在点z=z₁收敛,则容易证明,对每一z₁∣z∣<∣z₁∣ ,级数绝对收敛;

2. 因为∑a_nzⁿ在z=z₁收敛,于是,由weierstrass的M—判别法可立即得到∑a_nzⁿ在点z,∣z∣<z₁的绝对收敛性;

3. 我们知道有限项和中各项可以重新安排而不影响和的值,但对于无穷级数,上述结论却不总是真的

作者: hehe123 时间: 2004-11-27 12:47
标题: 数学专业英语[7]－Linear Algebra

For the definition that follows we assume that we are given a particular field K. The scalars to be used are to be elements of K.

DEFINITION. A vector space is a set V of elements called vectors satisfying the following axioms.

(A) To every pair, x and y ,of vectors in V corresponds a vector x+y,called the sum of x and y, in such a way that.

(1) addition is commutative, x + y = y + x.

(2) addition is associative, x + ( y + z ) = ( x + y ) + z.

(3) there exists in V a unique vector 0 (called the origin ) such that x + 0 = x for every vector x , and

(4) to every vector x in V there corresponds a unique vector - x such that x + ( - x ) = 0.

(B) To every pair,αand x , where α is a scalar and x is a vector in V ,there corresponds a vector αx in V , called the product of α and x , in such a way that

(1) multiplication by scalars is associative,α(βx ) = (αβ) x

(2) 1 x = x for every vector x.

(2)multiplication by vectors is distributive with respect to scalar addition,(α+β) x = αx + βx .

The relation between a vector space V and the underlying field K is usually described by saying that V is a vector space over K . The associated field of scalars is usually either the real numbers R or the complex numbers C . If V is linear space and M真包含于V , and if α u -v belong to M for every u and v in M and every α∈ K , then M is linear subspace of V . If U = { u 1,u 2,…} is a collection of points in a linear space V , then the (linear) span of the set U is the set of all points o the form ∑ c _i u _i , where c _i∈ K ,and all but a finite number of the scalars c_i are 0.The span of U is always a linear subspace of V.

A key concept in linear algebra is independence. A finite set { u ₁,u ₂,…, u _k} is said to be linearly independent in V if the only way to write 0 = ∑ c _i u _i is by choosing all the c _i = 0 . An infinite set is linearly independent if every finite set is independent . If a set is not independent, it is linearly dependent, and in this case, some point in the set can be written as a linear combination of other points in the set. A basis for a linear space M is an independent set that spans M . A space M is finite-dimensional if it can be spanned by a finite set; it can then be shown that every spanning set contains a basis, and every basis for M has the same number of points in it. This common number is called the dimension of M .

Another key concept is that of linear transformation. If V and W are linear spaces with the same scalar field K , a mapping L from V into W is called linear if L (u + v ) = L( u ) + L ( v ) and L ( αu ) = α L ( u ) for every u and v in V and α in K . With any I , are associated two special linear spaces:

ker ( L ) = null space of L = L^-1(0)

= { all x ∈ V such that L ( X ) = 0 }

Im ( L ) = image of L = L( V ) = { all L( x ) for x∈ V }.

Then r = dimension of Im ( L ) is called the rank of L. If W also has dimension n, then the following useful criterion results: L is 1-to-1 if and only if L is onto.In particular, if L is a linear map of V into itself, and the only solution of L( x ) = 0 is 0, then L IS onto and is therefore an isomorphism of V onto V , and has an inverse L ^-1 . Such a transformation V is also said to be nonsingular.

Suppose now that L is a linear transformation from V into W where dim ( V ) = n and dim ( W ) = m . Choose a basis {υ₁,υ_{2 ,}…,υ_n} for V and a basis {w ₁,w₂,…,w _m} for W . Then these define isomorphisms of V onto Kⁿand W onto K^m , respectively, and these in turn induce a linear transformation A between these. Any linear transformation ( such as A ) between Kⁿand K^mis described by means of a matrix ( a_ij), according to the formula A ( x ) = y , where x = { x₁ , x ₂,…, x_n } y = { y₁ , y ₂,…, y _m} and

Y _j =Σⁿ_j=i a_ij x_i I=1,2,…,m.

The matrix A is said to represent the transformation L and to be the representation induced by the particular basis chosen for V and W .

If S and T are linear transformations of V into itself, so is the compositic transformation ST . If we choose a basis in V , and use this to obtain matrix representations for these, with A representing S and B representing T , then ST must have a matrix representation C . This is defined to be the product AB of the matrixes A and B , and leads to the standard formula for matrix multiplication.

The least satisfactory aspect of linear algebra is still the theory of determinants even though this is the most ancient portion of the theory, dating back to Leibniz if not to early China. One standard approach to determinants is to regard an n -by- n matrix as an ordered array of vectors( u ₁, u ₂ ,…, u _n ) and then its determinant det ( A ) as a function F( u ₁, u ₂,…, u _n ) of these n vectors which obeys certain rules.

The determinant of such an array A turns out to be a convenient criterion for characterizing the nonsingularity of the associated linear transformation, since det ( A ) = F ( u ₁ , u ₂ ,…, u _n ) = 0 if and only if the set of vectors u_iare linearly dependent. There are many other useful and elegant properties of determinants, most of which will be found in any classic book on linear algebra. Thus, det ( AB ) = det ( A ) det ( B ), and det ( A ) = det ( A') ,where A' is the transpose of A , obtained by the formula A' =( a _ji), thereby rotating the array about the main diagonal. If a square matrix is triangular, meaning that all its entries above the main diagonal are 0,then det ( A ) turns out to be exactly the product of the diagonal entries.

Another useful concept is that of eigenvalue. A scalar is said to be an eigenvalue for a transformation T if there is a nonzero vector υ with T (υ) λυ . It is then clear that the eigenvalues will be those numbers λ∈ K such that T -λ I is a singular transformation. Any vector in the null space of T -λ I is called an eigenvector of T associated with eigenvalue λ, and their span the eigenspace, E _λ. It is invariant under the action of T , meaning that T carries E_λ into itself. The eigenvalues of T are then exactly the set of roots of the polynomial p(λ) =det ( T -λ I ).If A is a matrix representing T ,then one has p (λ) det ( A -λI ), which permits one to find the eigenvalues of T easily if the dimension of V is not too large, or if the matrix A is simple enough. The eigenvalues and eigenspaces of T provide a means by which the nature and structure of the linear transformation T can be examined in detail.

作者: hehe123 时间: 2004-11-27 12:48

Vocabulary

. L3 x( b3 l; g

linear algebra 线性代数 non-singular 非奇异

field 域 isomorphism 同构

vector 向量 isomorphic 同构

scalar 纯量,无向量 matrix 矩阵(单数)

vector space 向量空间 matrices 矩阵(多数)

span 生成,长成 determinant 行列式

independence 无关(性),独立(性) array 阵列

dependence 有关(性) diagonal 对角线

linear combination 线性组合 triangular 三角形的

basis 基(单数) entry 表值,元素

basis 基(多数) eigenvalue 特征值,本征值

dimension 维 eigenvector 特征向量

linear transformation 线性变换 invariant 不变,不变量

null space 零空间 row 行

rank 秩 column 列

singular 奇异 system of equations 方程组

homogeneous 齐次

作者: hehe123 时间: 2004-11-27 12:48

Notes

1. If U = { u ₁ , u ₂ ,…}is a collection of points in a linear

space V , then the (linear) span of the set U is the set of all points of the form ∑ c_i u_{i , w}where c _i∈ K ,and all but a finite number of scalars c_Iare 0.

意思是:如果 U = { u ₁ , u ₂ ,…}是线性空间 V 的点集,那么集 U 的(线性)生成是所有形如 ∑ c _i u _i 的点集,这里 c _i∈ K ,且除了有限个 c_i 外均为0.

& Q# {* E: P3 _# J3 ^/ { |

2. A finite set { u ₁ , u ₂ ,…, u _k}is said to be linearly independent if the only way to write 0 = ∑ c _i u _I is by choosing all the c _i= 0.

这一句可以用更典型的句子表达如下: A finite set { u ₁, u ₂ ,…, u _k } is said to be linearly independent in V if ∑c _i u _i is by choosing all the c _i = 0.

这里independent 是形容词,故用linearly修饰它. 试比较F(x) is a continuous periodic function.这里periodic 是形容词但它前面的词却用continuous 而不用continuously,这是因为continuous 这个词不是修饰periodic而是修饰作为整体的名词periodic function.

3. Then these define isomorphisms of V onto Kⁿ and W onto K^M respectively, and these in turn induce a linear transformation A between these.

这里第一个these代表前句的两个基(basis);第二个these代表isomorphisms;第三个these代表什么留给读者自己分析.

4. The least satisfactory aspect of linear algebra is still the theory of determinants-

意思是:线性代数最令人不满意的方面仍是有关行列式的理论.least satisfactory 意思是:最令人不满意.

5. If a square matrix is triangular, meaning that all its entries above the main diagonal are 0,then det ( A ) turns out to be exactly the product of the diagonal entries.

意思是:如果方阵是三角形的,即所有在主对角线上方的元素均为零,那末det( A ) 刚好就是对角线元素的乘积.这里meaning that 可用that is to say 代替,turns out to be解为”结果是

作者: hehe123 时间: 2004-11-27 12:49

Exercise

I. Answer the following questions:

1. How can we define the linear independence of an infinite set?

2. Let T be a linear transformation (T: V → W ) whose associated matrix is A.Give a criterion for the non-singularity of the transformation T.

3. Where is the entry a₄₅ of a m -by- n matrix( m>4; n>5) located ?

4. Let A , B be two rectangular matrices.Under what condition is the product matrix well-defined ?

0 S% l @7 f6 z1 z

II.Translate the following two examples and their proofs into Chinese:

1.Example1. Let u_k= t^k ,k=0,1,2,... and t real. Show that the set ｛u ₀,u₁,u₂,…｝ is independent.

Proof: By the definition of independence of an infinite set, it suffices to show that for each n ,the n+1 polynomials u₀,u₁,...,u_n are independent.A relation of the form ∑ⁿ_k=0 c_ku_k=0 means ∑ⁿ_k=0 c_k t^k=0 for all t.When t=0,this gives c₀ =0.Differentiating both sides of ∑ⁿ_k=0 c_k t^k =0 and setting t=0,we find that c₁=0.Repeating the process,we find that each cocfficient is zero

2. Example 2. Let V be afinite dimensional linear space, Then every finite basis for V has the same number of elements.

Proof: Let S and T be two finite bases for V. Suppose S consists of k elemnts and T consists of m elements.Since S is independent and spans V ,every set of k+1 elements in V is dependent.Therefore every set of more than k elements in V is dependent. Since T is an independent set , we must have m<k. The same argument with S and T interchanged shows that k<m. Hence k=m.

III.Translate the following sentences into English:

1.设 A 是一矩阵。若其行数 n 等于其列数 m ，则称 A 是一方阵；若n ≠m，则称A 是一矩形阵。

2.设 T 是一线性变换，则 T 的特征值刚好是多项式 P（λ）=det（T- λE ）的根。

3.形如Σⁿ_k=1c_ik x_k =c_i i=1，2，...,m 的一组方程称为 m 个线性方程，n 个未知数的方程组。若所有 c_i=0，则称上述方程组为一齐次方程组。

作者: hehe123 时间: 2004-11-27 12:49
标题: 数学专业英语[8]－Statistics
数学专业英语－Statistics/ E; j: A4 S0 T" ?' l& t

The term statistics is used in either of two senses.In common parlance it is generally employed synonymously with the word data.Thus someone may say that he has seen”statistics of industrial accidents in the United States.” It would be conducive to greater precision of meaning if we were not to use statistics in this sense,but rather to say “data (or figures ) of industrial accidents in the United States.”

“Statistics” also refers to the statistical principles and methods which have been developed for handling numerical data and which form the subject matter of this text.Statistical methods,or statistics, range form the most elementary descriptive devices, which may be understood by anyone , to those extremely complicated mathematical procedures which are comprehended by only the most expert theoreticians.It is the purpose of this volume not to enter into the highly mathematical and theoretical aspects of the subject but rather to treat of its more elementary and more frequently used phases.

Statistics may be defined as the collection, presentation, analysis, and interpretation of numerical data.The facts which are dealt with must be capable of numerical expression.We can make little use statistically of the information that dwellings are built of brick, stone, wood, and other materials; however, if we are able to determine how many or what proportion of,dwellings are constructed of each type of material, we have numerical data suitable for statistical analysis.

Statistics should not be thought of as a subject correlative with physics, chemistry, economics, and sociology. Statistics is not a science; it is a scientific method. The methods and procedures which we are about to examine constitute a useful and often indispensable tool for the research worker. Without an adequate understanding of statistics, the investigator in the social sciences may frequently be like the blind man groping in a dark closet for a black cat that isn’t there. The methods of statistics are useful in an ever---widening range of human activities, in any field of thought in which numerical data may be had.

In defining statistics it was pointed out that the numerical data are collected, presented, analyzed, and interpreted. Let us briefly examine each of these four procedures.

COLLECTION Statistical data may be obtained from existing published or unpublished sources, such as government agencies, trade associations, research bureaus, magazines, newspapers, individual research workers, and elsewhere. On the other hand, the investigator may collect his own information, going perhaps from house to house or from firm to firm to obtain his data. The first-hand collection of statistical data is one of the most difficult and important tasks which a statistician must face. The soundness of his procedure determines in an overwhelming degree the usefulness of the data which he obtains.

It should be emphasized, however, that the investigator who has experience and good common sense is at a distinct advantage if original data must be collected. There is much which may be taught about this phase of statistics, but there is much more which can be learned only through experience. Although a person may never collect statistical data for his own use and may always use published sources, it is essential that he have a working knowledge of the processes of collection and that he be able to evaluate the reliability of the data he proposes to use. Untrustworthy data do not constitute a satisfactory base upon which to rest a conclusion.

It is to be regretted that many people have a tendency to accept statistical data without question. To them, any statement which is presented in numerical terms is correct and its authenticity is automatically established.

PRESENTATION Either for one’s own use or for the use of others, the data must be presented in some suitable form. Usually the figures are arranged in tables or presented by graphic devices.

ANALYSIS In the process of analysis, data must be classified into useful and logical categories. The possible categories must be considered when plans are made for collecting the data, and the data must be classified as they are tabulated and before they can be shown graphically. Thus the process of analysis is partially concurrent with collection and presentation.

There are four important bases of classification of statistical data: (1) qualitative, (2) quantitative, (3) chronological, and (4) geographical, each of which will be examined in turn.

Qualitative When, for example, employees are classified as union or non—union, we have a qualitative differentiation. The distinction is one of kind rather than of amount. Individuals may be classified concerning marital status, as single, married, widowed, divorced, and separated. Farm operators may be classified as full owners, part owners, managers, and tenants. Natural rubber may be designated as plantation or wild according to its source.

Quantitative When items vary in respect to some measurable characteristics, a quantitative classification is appropriate. Families may be classified according to the number of children. Manufacturing concerns may be classified according to the number of workers employed, and also according to the values of goods produced. Individuals may be classified according to the amount of income tax paid.

Chronological Chronological data or time series show figures concerning a particular phenomenon at various specified times. For example, the closing price of a certain stock may be shown for each day over a period of months of years; the birth rate in the United States may be listed for each of a number of years; production of coal may be shown monthly for a span of years. The analysis of time series, involving a consideration of trend, cyclical period (seasonal ), and irregular movements, will be discussed.

In a certain sense, time series are somewhat akin to quantitative distributions in that each succeeding year or month of a series is one year or one month further removed from some earlier point of reference. However, periods of time—or, rather, the events occurring within these periods—differ qualitatively from each other also. The essential arrangement of the figures in a time sequence is inherent in the nature of the data under consideration.

Geographical The geographical distribution is essentially a type of qualitative distribution, but is generally considered as a distinct classification. When the population is shown for each of the states in the United States, we have data which are classified geographically. Although there is a qualitative difference between any two states, the distinction that is being made is not so much of kind as of location.

The presentation of classified data in tabular and graphic form is but one elementary step in the analysis of statistical data. Many other processes are described in the following passages of this book. Statistical investigation frequently endeavors to ascertain what is typical in a given situation. Hence all type of occurrences must be considered, both the usual and the unusual.

In forming an opinion, most individuals are apt to be unduly influenced by unusual occurrences and to disregard the ordinary happenings. In any sort or investigation, statistical or otherwise, the unusual cases must not exert undue influence. Many people are of the opinion that to break a mirror brings bad luck. Having broken a mirror, a person is apt to be on the lookout for the unexpected”bad luck “ and to attribute any untoward event to the breaking of the mirror. If nothing happens after the mirror has been broken, there is nothing to remember and this result (perhaps the usual result )is disregarded. If bad luck occurs, it is so unusual that it is remembered, and consequently the belief is reinforced. The scienticfic procedure would include all happenings following the breaking of the mirror, and would compare the “resulting” bad luck to the amount of bad luck occurring when a mirror has not been broken.

Statistics, then, must include in its analysis all sorts of happenings. If we are studying the duration of cases of pncumonia, we may study what is typical by determining the average length and possibly also the divergence below and above the average. When considering a time series showing steel—mill activity, we may give attention to the typical seasonal pattern of the series, to the growth factor( trend) present, and to the cyclical behaviour. Sometimes it is found that two sets of statistical data tend to be associated.

Occasionlly a statistical investigation may be exhaustive and include all possible occurrences. More frequently, however, it is necessary to study a small group or sample. If we desire to study the expenditures of lawyers for life insurance, it would hardly be possible to include all lawyers in the United States. Resort must be had to a sample;and it is essential that the sample be as nearly representative as possible of the entire group, so that we may be able to make a reasonable inference as to the results to be expected for an entire population. The problem of selecting a sample is discussed in the following chapter.

Sometimes the statistician is faced with the task of forecasting. He may be required to prognosticate the sales of automobile tires a year hence, or to forecast the population some years in advance. Several years ago a student appeared in summer session class of one of the writers. In a private talk he announced that he had come to the course for a single purpose: to get a formula which would enable him to forecast the price of cotton. It was important to him and his employers to have some advance information on cotton prices, since the concern purchased enormous quantities of cotton. Regrettably, the young man had to be disillusioned. To our knowledge, there are no magic formulae for forecasting. This does not mean that forecasting is impossible; rather it means that forecasting is a complicated process of which a formula is but a small part. And forecasting is uncertain and dangerous. To attempt to say what will happen in the future requires a thorough grasp of the subject to be forecast, up-to-the-minute knowledge of developments in allied fields, and recognition of the limitations of any mechanica forecasting device.

INTERPRETATION The final step in an investigation consists of interpreting the data which have been obtained. What are the conclusions growing out of the analysis? What do the figures tell us that is new or that reinforces or casts doubt upon previous hypotheses? The results must be interpreted in the light of the limitations of the original material. Too exact conclusions must not be drawn from data which themselves are but approximations. It is essential, however, that the investigator discover and clarify all the useful and applicable meaning which is present in his data.

5 j$ G: b: `2 @/ M4 j# {- f( n* _3 M1 ~& ^% S, C$ W' P: F ' i. j' g0 P+ M' J

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2 G. Z2 p: V2 |4 ]

! W" f# R1 ` t7 D3 C. C; p) r. k% O

作者: hehe123 时间: 2004-11-27 12:50

Vocabulary

Statistics统计学 in tables 列成表

Statistical 统计的 tabular列成表的

Statistical data统计数据 sample样本

Statistical method统计方法 inference推理，推断

Original data原始数据 reliance信赖

Qualitative定性的 forecasting预测

Quantitative定量的 in common parlance按一般说法

Chronological年代学的 conducive有帮助的

Time series时间序列 grope摸索

Cyclical循环的 akin to类似

Period周期 apt to易于

Periodic周期的 undue不适当的

Prognosticate预测 sociology社会学

Authenticity可信性，真实性 phase相位；方面

Synonymously同义的 categories范畴，类型

Correlative相互关系的，相依的 concurrent会合的，一致的，同时发生的

作者: hehe123 时间: 2004-11-27 12:50

Notes 5 Y& w; Y; \4 i

1. It is the purpose of this volume not to enter into the highly mathematical and theoretical aspects of the subject but rather to treat of its more elementary and more frequently used phases.

意思是：本书的目的并不是要深入到这个论题的有关高深的数学与理论的方面，而是要讨论它的更为初等和更为常用的方面，not…but rather 意思是“不是…而是”，而rather than意思是“宁愿…而不”，两者意思相近但有差别（主要表现为强调哪方面的差别）。

2. Without an adequate understanding of statistics, the investigator in the social sciences may frequently be like the blind man groping in a dark closet for a black cat that isn’t there.

意思是：没有对统计学的适当的理解，社会科学的研究人员就会常常像一个盲人在一个暗室里摸索一只并不存在的黑猫一样，这里investigator 指一般研究人员，请比较与researcher，scientist等词的异同。

3. It is essential that he have a working knowledge of the processes of collection and that he be able…

这里essential是虚拟形容词，所以后面用he have；he be able…have和be前面省去了should。

4. The distinction is one of kind rather than of amount.

意思是：这种区别是类型的而不是数量的。

5. …the distinction that is being made is not so much of kind as of location.

参看第二课注2关于not so much…as的解释，并比较注4和注5两句。

6. （To attempt to say what will happen in the future） requires…of any mechanical forecasting device.

括号里的部分为此句的主语，谓语是requires，后面是三个并列宾语短语。

作者: hehe123 时间: 2004-11-27 12:51

Exercise

I. Translate the following passages into Chinese:

1. Statistical methods, or statistics, range from the most elementary descriptive devices which may be understood by anyone, to those extremely complicated mathematical procedures which are comprehended by only the most expert theoreticians.

2. Although a person may never collect statistical data for his own use and may always use published sources, it is essential that he have a working knowledge of the processes of collection and that he be able to evaluate the reliability of the data he proposes to use.

II Answer the following questions according to the content of this lesson.

1. In defining statistics, what are the four procedures about the numerical data?

2. How are the numerical data presented usually?

3. Give examples for qualitative differentiation and quantitative classification of statistical data .

' r) T( W7 U+ G

作者: hehe123 时间: 2004-11-27 12:51
标题: 数学专业英语[9]－Linear Programming

Linear Programming is a relatively new branch of mathematics.The cornerstone of this exciting field was laid independently bu Leonid V. Kantorovich,a Russian mathematician,and by Tjalling C,Koopmans, a Yale economist,and George D. Dantzig,a Stanford mathematician. Kantorovich’s pioneering work was motivated by a production-scheduling problem suggested by the Central Laboratory of the Leningrad Plywood Trust in the late 1930’s. The development in the United States was influenced by the scientific need in World War II to solve logistic military problems, such as deploying aircraft and submarines at strategic positions and airlifting supplies and personnel.

The following is a typical linear programming problem:

A manufacturing company makes two types of television sets: one is black and white and the other is color. The company has resources to make at most 300 sets a week. It takes $180 to make a black and white set and $270 to make a color set. The company does not want to spend more than $64,800 a week to make television sets. If they make a profit of $170 per black and white set and $225 per color set, how many sets of each type should the company make to have a maximum profit?

This problem is discussed in detail in Supplementary Reading Material Lesson 14.

Since mathematical models in linear programming problems consist of linear inequalities, the next section is devoted to such inequalities.

Recall that the linear equation lx+my+n=0 represents a straight line in a plane. Every solution (x,y) of the equation lx+my+n=0 is a point on this line, and vice versa.

An inequality that is obtained from the linear equation lx+my+n=0 by replacing the equality sign “=” by an inequality sign < (less than), ≤ (less than or equal to), > (greater than), or ≥ (greater than or equal to) is called a linear inequality in two variables x and y. Thus lx+my+n≤0, lx+my+n≥0 are all linear liequalities. A solution of a linear inequality is an ordered pair (x,y) of numbers x and y for which the inequality is true.

EXAMPLE 1 Graph the solution set of the pair of inequalities

SOLUTION Let A be the solution set of the inequality x+y-7≤0 and B be that of the inequality x-3y +6 ≥0 .Then A∩B is the solution set of the given pair of inequalities. Set A is represented by the region shaded with horizontal lines and set B by the region shaded with vertical lines in Fig.1. Therefore the crossed-hatched region represents the solution set of the given pair of inequalities. Observe that the point of intersection (3.4) of the two lines is in the solution set.

Generally speaking, linear programming problems consist of finding the maximum value or minimum value of a linear function, called the objective function, subject to some linear conditions, called constraints. For example, we may want to maximize the production or profit of a company or to maximize the number of airplanes that can land at or take off from an airport during peak hours; or we may want to minimize the cost of production or of transportation or to minimize grocery expenses while still meeting the recommended nutritional requirements, all subject to certain restrictions. Linear programming is a very useful tool that can effectively be applied to solve problems of this kind, as illustrated by the following example.

EXAMPLE 2 Maximize the function f(x,y)=5x+7y subject to the constraints

x≥0 y≥0

x+y-7≤0

2x-3y+6≥0

SOLUTION First we find the set of all possible pairs(x,y) of numbers that satisfy all four inequalities. Such a solution is called a feasible sulution of the problem. For example, (0,0) is a feasible solution since (0,0) satisfies the given conditions; so are (1,2) and (4,3). : k0 d! K/ O: y1 @6 d 4 V7 }# c( V( l 8 h" \7 Q9 W8 k* E1 {, ]' p

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Secondly, we want to pick the feasible solution for which the given function f (x,y) is a maximum or minimum (maximum in this case). Such a feasible solution is called an optimal solution. ! H- S2 h7 k/ [/ Z/ u

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6 {! @- g0 S! B, b' H( p( c) u! X4 g" L0 A

Since the constraints x ≥0 and y ≥0 restrict us to the first quadrant, it follows from example 1 that the given constraints define the polygonal region bounded by the lines x=0, y=0,x+y-7=0, and 2x-3y+6=0, as shown in Fig.2.

Fig.2.

Observe that if there are no conditions on the values of x and y, then the function f can take on any desired value. But recall that our goal is to determine the largest value of f (x,y)=5x+7y where the values of x and y are restricted by the given constraints: that is, we must locate that point (x,y) in the polygonal region OABC at which the expression 5x+7y has the maximum possible value.

With this in mind, let us consider the equation 5x+7y=C, where C is any number. This equation represents a family of parallel lines. Several members of this family, corresponding to different values of C, are exhibited in Fig.3. Notice that as the line 5x+7y=C moves up through the polygonal region OABC, the value of C increases steadily. It follows from the figure that the line 5x+7y=43 has a singular position in the family of lines 5x+7y=C. It is the line farthest from the origin that still passes through the set of feasible solutions. It yields the largest value of C: 43.(Remember, we are not interested in what happens outside the region OABC) Thus the largest value of the function f(x,y)=5x+7y subject to the condition that the point (x,y) must belong to the region OABC is 43; clearly this maximum value occurs at the point B(3,4).

Fig.3.

Consider the polygonal region OABC in Fig.3. This shaded region has the property that the line segment PQ joining any two points P and Q in the region lies entirely within the region. Such a set of points in a plane is called a convex set. An interesting observation about example 2 is that the maximum value of the objective function f occurs at a corner point of the polygonal convex set OABC, the point B(3,4).

The following celebrated theorem indicates that it was not accidental.

THEOREM (Fundamental theorem of linear programming) A linear objective function f defined over a polygonal convex set attains a maximum (or minimum) value at a corner point of the set.

We now summarize the procedure for solving a linear programming problem:

1. Graph the polygonal region determined by the constraints.

2. Find the coordinates of the corner points of the polygon.

3. Evaluate the objective function at the corner points.

4. Identify the corner point at which the function has an optimal value.

作者: hehe123 时间: 2004-11-27 12:52

Vocabulary

linear programming 线形规划 quadrant 象限

objective function 目标函数 convex 凸的

constraints 限制条件，约束条件 convex set 凸集

feaseble solution 容许解，可行解 corner point 偶角点

optimal solution 最优解 simplex method 单纯形法

作者: hehe123 时间: 2004-11-27 12:52

Notes

1. A Yale economist, a Stanford mathematician 这里Yale Stanford 是指美国两间著名的私立大学：耶鲁大学和斯坦福大学，这两间大学分别位于康涅狄格州（Connecticut）和加里福尼亚州(California)

2. subject to some lincar conditions 解作“在某些线形条件的限制下”。试比较下面各词组在用法上的异同： subject to; under the condition(s) of; satisfying the condition(s).

3. celebrated theorem 意思为“著名定理”。

4. Since…it follows…与Notice…it follow…都是数学中常用的句型，以表达“根据什么，可得什么”这一意思，请参看附录III。

5. 本课多次用到recall, observe, notice, remember等词，用以提醒读者一些已知的事实或定理，读者可从这些例句中体会这些词的用法。请参看附录III。

作者: hehe123 时间: 2004-11-27 12:53

Exercise

I.Translate the following passage into Chinesre:

Although satisfactory for two-variable linear programs, the corner-point method is not computationally effective when extended beyond three-variable situations. Its usefulness is restricted to introducing the general idea of linear programming, and the need for more poweful solution techniques is obvious. The simplex method, devised by George Dantzig in the late 1940’s, was, central to the explosion in linear programming applications that occurred in the 1950’s and 1960’s. Although more powerful techniques exist for solving special problems, the simplex method is still the most computationally effective general method available for solving the widest variety of linear programming problems.

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II.Read the content of this lesson carefully and complete the following sentences:

1. Linear programming problems consist of * F4 G- l. q1 P) G

* {$ H! s! C1 I* f! s$ B' r

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2. The set of feasible solutions of a linear programming problem is

3. A polygonal region is a region bounded by

4. A convex set in a plane is " V% F- q: d9 `, H

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III.Translate the following sentences into English:

1. 点(3,4)是两条直线x+y-7=0和2x-3y+6=0的交点。

2. (3,4)是例2的最优解。

3. 计算目标函数在偶角点处的值，然后进行比较，求出目标函数的最大值或最小值。

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作者: hehe123 时间: 2004-11-27 12:54
标题: 数学专业英语[10]－Groups and Rings

数学专业英语－Groups and Rings

During the present century modern abstract algebra has become more and more important as a tool for research not only in other branches of mathematics but even in other sciences .Many discoveries in abstract algebra itself have been made during the past years and the spirit of algebraic research has definitely tended toward more abstraction and rigor so as to obtain a theory of greatest possible generality. In particular, the concepts of group ,ring,integral domain and field have been emphasized.

The notion of an abstract group is fundamental in all sciences ,and it is certainly proper to begin our subject with this concept. Commutative additive groups are made into rings by assuming closure with respect to a second operation having some of the properties of ordinary multiplication. Integral domains and fields are rings restricted in special ways and may be fundamental concepts and their more elementary properties are the basis for modern algebra.

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Groups % c6 n3 y2 Z% m( D$ N

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DEFINITION A non-empty set G of elements a,b,…is said to form a group with respect to 0 if:

I. G is closed with respect to 0

II. The associative law holds in G, that is

aо(bоc)=(aоb)оc

for every a, b, c of G

Ⅲ. For every a and b of G there exist solutions χ and У in G of the equations

aοχ=b yοa=b

A group is thus a system consisting of a set of elements and operation ο with respect to which G forms a group. We shall generally designate the entire system by the set G of its elements and shall call G a group. The notation used for the operation is generally unimportant and may be taken in as convenient a way as possible.

DEFINITION A group G is called commutative or abelian if

aοb=bοa " i- Y9 D; U! P( G, c- M; U4 M

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An elementary physical example of an abelian group is a certain rotation group. We let G consist of the rotations of the spoke of a wheel through multiples of 90º and aοb be the result of the rotation a followed by the rotation b. The reader will easily verify that G forms a group with respect to ο and that aοb=bοa. There is no loss of generality when restrict our attention to multiplicative groups, that is, write ab in stead of aοb.

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EQUIVALENCE 9 ~% @* {8 k+ {

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In any study of mathematical systems the concept of equivalence of systems of the same kind always arises. Equivalent systems are logically distinct but we usually can replace any one by any other in a mathematical discussion with no loss of generality. For groups this notion is given by the definition: let G and G´ be groups with respective operations o and o´,and let there be a1-1 correspondence

S : a a´ (a in G and a´ in G´) 3 L. K) @, U* C

9 \# h4 d2 c, N/ R7 I7 |

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between G and G´ such that

(aοb)´=a´ο b´

for all a, b of G. then we call G and G´equivalent(or simply, isomorphic)groups.

The relation of equivalence is an equivalence relation in the technical sense in the set of all groups. We again emphasize that while equivalent groups may be logically distinct they have identical properties.

The groups G and G´ of the above definition need not be distinct of course and o´ may be o. when this is the case the self-equivalence S of G is called an automorphism.

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Of G, but other automorphisms may also exist.

Rings

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A ring is an additive abelian group B such that

I. the set B is closed with respect to a second operation designated by multiplication; that is , every a and b of B define a unique element ab of B.

II. multiplication is associative; that is

a (bc) = (ab)c

for every a, b, c of B.

Ⅲ. The distributive laws

a (b+c) = ab +ac (b+c) a=ba +ca 1 s1 ^/ z# P# f7 h n

6 h* p e# y: \, T

( L' L: ~: E# [, o; e7 i" x+ H6 `) u) @

hold for every a, b, c of B.

The concept of equivalence again arises. We shall write

B ≌ B′ ( R ]: U6 I6 H; G Y: J3 d: k% Y" h

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to mean that B and B′are equivalent.

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作者: hehe123 时间: 2004-11-27 12:54

Vocabulary " z- z! @4 E4 Q( r4 k

Group 群 rigor 严格

ring 环 generalization 推广

integral domain 整环 Abelian group 阿贝尔群

commutative additive group 可交换加法群 rotation 旋转

automorphism 自同构

作者: hehe123 时间: 2004-11-27 12:55
标题: 数学专业英语[11]－Historical introduction of Calculus

The Two Basic Concepts of Calculus 8 \, ?4 U- G5 l' D/ s' `9 `/ n+ O# D$ [$ x8 x3 ^( A : C# Y3 I2 U, t' f

! o' p4 y# C# Q1 ^& W, A

The remarkable progress that has been made in science and technology during the last century is due in large part to the development of mathematics. That branch of mathematics known as integral and differential calculus serves as a natural and powerful tool for attacking a variety of problems that arise in physics,engineering,chemistry,geology,biology, and other fields including,rather recently,some of the social sciences.

To give the reader an idea of the many different types of problems that can be treatedby the methods of calculus,we list here a few sample questions.

With what speed should a rocket be fired upward so that it never returns to earth? What is the radius of the smallest circular disk that can cover every isosceles triangle of a given perimeter Ｌ? What volume of material is removed from a solid sphere of radius 2 r if a hole of redius r is drilled through the center? If a strain of bacteria grows at a rate proportional to the amount present and if the population doubles in one hour,by how much will it increase at the end of two hours? If a ten-pound force stretches an elastic spring one inch,how much work is required to stretch the spring one foot?

These examples,chosen from various fields,illustrate some of the technical questions that can be answered by more or less routine applications of calculus.

Calculus is more than a technical tool－it is a collection of fascinating and exeiting idea that have interested thinking men for centuries.These ideas have to do with speed,area,volume,rate of growth,continuity,tangent line,and otherconcepts from a varicty of fields.Calculus forces us to stop and think carefully about the meanings of these concepts. Another remarkable feature of the subject is its unifying power.Most of these ideas can be formulated so that they revolve around two rather specialized problems of a geometric nature.We turn now to a brief description of these problems.

Consider a cruve C which lies above a horizontal base line such as that shown in Fig.1. We assume this curve has the property that every vertical line intersects it once at most.The shaded portion of the figure consists of those pointe which lie below the curve C , above the horizontal base,and between two parallel vertical segments joining C to the base.The first fundamental problem of calculus is this: To assign a number which measures the area of this shaded region.

Consider next a line drawn tangent to the curve,as shown in Fig.1. The second fundamental problem may be stated as follows:To assign a number which measures the steepness of this line.

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Basically,calculus has to do with the precise formulation and solution of these two special problems.It enables us to define the concepts of area and tangent line and to calculate the area of a given region or the steepness of a given angent line. Integral calculus deals with the problem of area while differential calculus deals with the problem of tangents.

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Historical Background " t+ n b; Q! @( m) G; L. j

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The birth of integral calculus occurred more than 2000 years ago when the Greeks attempted to determine areas by a procees which they called the method of exhaustion.The essential ideas of this ,method are very simple and can be described briefly as follows:Given a region whose area is to be determined,we inscribe in it a polygonal region which approximates the given region and whose area we can easily compute.Then we choose another polygonal region which gives a better approximation,and we continue the process,taking polygons with more and more sides in an attempt to exhaust the given region.The method is illustrated for a scmicircular region in Fig.2. It was used successfully by Archimedes(287－212 B.C.) to find exact formulas for the area of a circle and a few other special figures.

The development of the method of exhaustion beyond the point to which Archimcdcs carried it had to wait nearly eighteen centuries until the use of algebraic symbols and techniques became a standard part of mathematics. The elementary algebra that is familiar to most high-school students today was completely unknown in Archimedes’ time,and it would have been next to impossible to extend his method to any general class of regions without some convenient way of expressing rather lengthy calculations in a compact and simpolified form.

A slow but revolutionary change in the development of mathematical notations began in the 16th century A.D. The cumbersome system of Roman numerals was gradually displaced by the Hindu-Arabic characters used today,the symbols “+”and “-”were introduced for the forst time,and the advantages of the decimal notation began to be recognized.During this same period,the brilliant successe of the Italian mathematicians Tartaglia,Cardano and Ferrari in finding algebraic solutions of cubic and quadratic equations stimulated a great deal of activity in mathematics and encouraged the growth and acceptance of a new and superior algebraic language. With the wide spread introduction of well-chosen algebraic symbols,interest was revived in the ancient method of exhaustion and a large number of fragmentary results were discovered in the 16 th century by such pioneers as Cavalieri, Toricelli, Roberval, Fermat, Pascal, and Wallis.

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2 Q+ L( X0 ]5 \: ]2 d, O& d) S

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Fig.2. The method of exhaustion applied to a semicircular region.

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Gradually the method of exhaustion was transformed into the subject now called integral calculus,a new and powerful discipline with a large variety of applications, not only to geometrical problems concerned with areas and volumes but also to jproblems in other sciences. This branch of mathematics, which retained some of the original features of the method of exhaustion,received its biggest impetus in the 17 th century, largely due to the efforts of Isaac Newion (1642—1727) and Gottfried Leibniz (1646—1716), and its development continued well into the 19 th century before the subject was put on a firm mathematical basis by such men as Augustin-Louis Cauchy (1789-1857) and Bernhard Riemann (1826-1866).Further refinements and extensions of the theory are still being carried out in contemporary mathematics.

作者: hehe123 时间: 2004-11-27 12:55

Vocabulary 4 Q8 |2 X/ u: h* c j

geology 地质学 decimal 小数，十进小数

biology 生物学 discipline 学科

social sciences 社会科学 contemporary 现代的

disk (disc) 圆盘 bacteria 细菌

isosceles triangle 等腰三角形 elastic 弹性的

perimeter 周长 impetus 动力

volume 体积 proportional to 与…成比例

center 中心 inscribe 内接

steepness 斜度 solid sphere 实心球

method of exhaustion 穷举法 refinement 精炼，提炼

polygon 多边形，多角形 cumbersome 笨重的，麻烦的

polygonal 多角形 fragmentary 碎片的，不完全的

approximation 近似，逼近 background 背景

作者: hehe123 时间: 2004-11-27 12:56
标题: 数学专业英语[12]－The Normal Distribution

We shall begin by considering some simple continuously variable quantity like stature.We know this varies greatly from one individual to another ,and may also expect to find certain average differences between people drawn from different social classes or living in different geographical areas,etc.Let us suppose that a socio-medical survey of a particular community has provided us with a representative sample of 117 males whose heights are distributed as shown in the first and third columns of Table 1.

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Table 1.Distribution of stature in 117 males

Absolute

Height

(m)

Working

measurements

with origin

at 1.70(x)

Number of

men

observed(f)

Contributions

to the sum

(f x )

Contributions

to the sum of

squares (f )

1.58

1.60

1.62

1.64

1.66

1.68

1.70

1.72

1.74

1.76

1.78

1.80

1.82

1.84

－6

－5

－4

－3

－2

－1

－6

－15

－24

－26

－18

100

Ｔotals

4 x- O: { V( H& k9 d

9 e' V+ O5 \" p# N2 a. C2 j& C

117

＋15

801

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We shall assume that the original measurements were made as accurately as possible,but that they are given here only to the mearest 0.02 m (i.e.2 cm).Thus the group labeled “1.66” contains all those men whose true measurements were between 1065 and 1067 m.One si biable to run into trouble if the exact methods of recording the measurements and grouping them are not specified exactly.In the example just given the mid-point of the interval labeled”1.66” m.But suppose that the original readings were made only to the nearest 0.01 m (i.e. 1 cm )and then “rounded up “to the nearest multiple of 0.02 m.We should then have “1.65”, which covers the range 1.645 to 1.655,included with “1.66”.The interval “1.66”would then contain all measurements lying between 1.645 m and 1.665 m .for which the mid-point is 1.655 m. The difference of 5 mm from the supposed value of 1.66 m could lead to serious inaccuracy in certain types of investigation.

A convenient visual way of presenting such data is shown in fig. 1, in which the area of each rectangle is ,on the scale used, equal to the observed proportion or percentage of individuals whose height falls in the corresponding group.The total area covered by all the rectangles therefore adds up to unity or 100per cent .This diagram is called a histogram.It is easily constructed when ,sa here ,all the groups are of the same width.It is also easily adapted to the case when the intervals are uneqal, provided we remember that the areas of the rectangles must be proportional to the numbers of units concerned.If, for example, we wished to group togcther the entries for the three groups 1.80,1.82 and 1.84 m,totaling 7 individuals or 6 per cent of the total,then we should need a rectangle whose base covered 3working groups on the horizontal scale but whose height was only 2 units on the vertical scale shown in the diagram.In this way we can make allowance for unequal grouping intervals ,but it is usually less troublesome if we can manageto keep them all the same width.In some books histograms are drawn so that the area of each rectangle is equal to the actual number (instend of the proportion) of individuals in the corresponding group.It is better, however, to use proportions, sa different histograms can then be compared directly.

作者: hehe123 时间: 2004-11-27 12:57

The general appearance of the rectangles in Fig.1 is quite striking ,especially the tall hump in the centre and the rapidly falling tails on each side.There are certain minor irregularities in the pattern, and these would, in general ,be more ronounced if the size of the sample were smaller. Conversely, weth larger samples we usually find that the set of rectangles presents a more regular appearance. This suggests that if we had a very large number of measurements ,the ultimate shape of the picture for a suitably small width of rectangle would be something very like a smooth curve,Such a curve could be regarded as representing the true ,theoretical or ideal distribution of heights in a very (or ,better,infinitely)large population of individuals.

What sort of ideal curve can we expect ? There are seveala theoretical reasons for expecting the so-called Gaussiao or “normal “curve to turn up in practice;and it is an empirical fact that such a curve lften describes with sufficient accuracy the shape of histograms based on large numbers of obscrvations. Moreover,the normal curve is one of the easiest to handle theoretically,and it leads to types of statistical analysis that can be carried out with a minimum amount of computation. Hence the central importance of this distribution in statistical work .

The actual mathematical equation of the normal curve is where u is the mean or average value and is the standard deviation, which is a measure of the concentration of frequency about the mean. More will be said about and later .The ideal variable x may take any value from to .However ,some real measurements,like stature, may be essentially positive. But if small values are very rare ,the ideal normal curve may be a sufficiently close approximation. Those readers who are anxious to avoid as much algebraic manipulation as possible can be reassured by the promise that no derect use will be made in this book of the equation shown. Most of the practical numerical calculations to which it leads are fairly simple.

Fig. 1 shows a normal curve, with its typocal symmetrical bell shape , fitted by suitable methods to the data embodied in the rectangles. This is not to say that the fitted curve is actually the true, ideal one to which the histogram approxime.tes; it is merely the best approximation we can find.

The mormal curve used above is the curve we have chosen to represent the frequency distribution of stature for thr ideal or infinitely large population. This ideal poplation should be contrasted with the limited sample of obsrever. Values that turns up on any occasion when we make actual measurements in the real world. In the survey mentioned above we had a sample of 117 men .If the community were sufficiently large for us to collect several samples of this size, we should find that few if any of the corresponding histograms were exactly the same ,although they might all be taken as illustrating the underlying frequency distribution. The differences between such histograms constitute what we call sampling variation, and this becomes more prominent at the size of sample decreases

作者: hehe123 时间: 2004-11-27 12:57

Vocabulary / ~7 H6 V6 j5 q6 c9 e _

Socio-medical survey 社会医疗调查表 visual 可见的。

distribute 分布（动词） percentage 百分比

distribution 分布（名词） individual 个人，个别

histogram 直方图，矩形图 mean 平均值，中数

hump 驼背，使隆起 standard deviation 标准差

normal distribution 正态分布 sample varianice 样本方差

作者: hehe123 时间: 2004-11-27 12:58
标题: 数学专业英语[13]－Operations Research

数学专业英语－Operations Research

The start of operations research took place in a military context in the United Kingdom during World War Ⅱ, and it was quickly taken up under the name operations research (OR) in the United States. After the war it evolved in connection with industrial organization, and its many techniques allowed for expanding areas of application in the United States, the United Kingdom, and in other industrial countries. It is, however, not easy to give a precise definition of operations research, There are three different representative definitions.

According to the classical definition, due to P. M. Morse and G. E. Kimball, operations research is a scientific method of providing executives with a quantitative basis for decisions regarding operations under their control.

The second definition, due to C. W. Churchman, R. L. Ackoff, and E. L. Arnoff, is as follows: operations research in the most general sense can be characterized as the application of scientific methods, techniques, and tools to the operations of systems so as to provide those in control with optimum solutions to problems.

As the third definition we mention the suggestion due to S. Beer: operations research is the attack of modern science on problems of likelihood (accepting mischance) that arise in the management and control of men, machines, materials, and money in their natural environments. Its special technique is to invent a strategy of control by measuring, comparing, and predicting probable behaviour through a scientific model of a situation.

These three definitions have several common features. In the first place, operations research serves executives by providing partial observations and advice which they can use in judging a situation. Second, the applicability of operations research is limited to areas where scientific methods can be successfully applied. This is the reason why operations research would not be considered to extend beyond only partial observation and advice. A fundamental requirement for a scientific approach is that it must have a mathematical model whose validity can be tested by actual data, Third, any operation should satisfy three necessary conditions in order that it may be an object of scientific approach: (1) the operation should be defined objectively; (2) the results, consequences, and effects of its application should be objectively measurable; (3) the operation should be capable of repetition. Fourth , operations research should aim at finding a practical strategy. Although operations research is based on scientific methodology, it does not aim at establishing general scientific assertions that are valid for all situstions.

These four points are essential to any operations research, and are implicit in each of the three aforementioned definitions.

On the other hand these three definitions emphasize differently some specific features of operations research, according to their historical positions. In comparion with the first definition, the second makes clearer the place where operations research is applied by pointing out that it is concerned with the operations of systems, and, instead of the vague mention of quantitative basis for decisions in the first definition, it states that operations research seeks optimum solutions, reflecting a stage where optimum solutions were sought by applications of mathematical programming techniques. In the third definition of operations research the notion of system is defined explicitly, the notion of operation is defined to be its special technique, and the objectives of operations research are given. It is clearly asserted that operations research belongs to the methodology of applied sciences. In operations research, operations and systems are dealt with in their intimate interconnection. The methodology of operations research therefore relies on an overall approach for which interdisciplinary cooperation is indispensable and in which the operations research team plays an important role.

In applying the operations research approach to the circumstances with which we are concerned, we concentrate our interest on mutual relationships among input and output characteristics. A black-box method by which the interrelation between input and output can be clarified without entering the actual mechanism of the transformation yielded by the system or by its subsystems plays a fundamental role in operations research. The following are major phases of an operations research project: (1) formulating the problem; (2) constructing a mathematical model to represent the system under study and deriving a solution from the model; (3) testing the model and the solution derived form it; (4) the implementation stage of establishing controls over the solution and putting it to work. It is important to construct a model of information communication in connection. With a mathematical model of any problem in operations research. Process of aliocation, competition, queuing, inventory, and production appear frequently in the mathematical models of operations research.

6 ]$ D% U! S z1 @2 p1 d u; H3 k; O' m$ |2 \ ) h, O# T3 G% p( T. M- K

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——From Encyclopedic Dictionary

of Mathematics

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作者: hehe123 时间: 2004-11-27 12:58

Vocabulary & ^# y4 s- ^7 a5 k

Operations research (OR) 运筹 interdiscipline 交叉学科

Executive 行政人员 interdisciplinary cooperation 交叉学科的

likelihood 似然合作

scientific approach 科学方法 black-box method 黑箱方法

methodology 方法论 implementation stage 实现阶段

aforementioned 前述的 queue 排队

mathematical programming 数学规划

作者: hehe123 时间: 2004-11-27 12:58

Notes

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1. Operations Research运筹学. 运筹学是第二次世界大战期间,为解决后勤供应问题而发展起来的一门学科,它运用最优化技术去解决管理和决策问题.

2. According to the classical definition, due to P. M. Morse and G. E. Kimball, operations research is a scientific method of providing executives with a quantitive basis for decisions regarding operations under their control.

意思是:根据P. M. Morse和G. E. Kimball提出的古典定义,运筹学是一种科学方法,它提供行政人员一种定量基础,以便他们对所控的操作进行决策,这里due to是”归功于” “由…提出”之意,providing…with…for…是”提供…给…用于…”之意.

3. …instead of the vague mention of quantitative basis for decisions in the first definition, it states that …by applications of mathematical programming techniques.

意思是:代替第意个定义中对于决策的定量基础那种模糊的提法,它(第二个定义)阐明了运筹学用于寻求最优解,反映了运用数学规划方法求最优解的阶段.这里reflecting至句子结束一段,属独立分词结构,用以补充说明it states that…的句子.

4. The methodology of operations research therefore relies on…the operations research team plays an important role.

意思是:因而运筹学的方法论依赖于…一种全面的研究,对这种研究来说,各交叉学科的合作是不可避免的,而且,在这种研究中,运筹学小组起了重要的作用.注意:前后两个which都是approach的关系代词,很容易误认为第二个which 是cooperation的关系代词,虽然这在意思上说得过去,但从语法结构上却不然.

5. A black-box method by which the interrelation between input and output …plays a fundamental role in operations research.

意思是:黑箱方法不需引进由系统或它的子系统所产生的变换的确切机制而能阐明输入和输出的相互关系,这种方法在运筹学中起了重要的作用,注意这一句中的主语A black-box and method和谓语plays相隔甚远.

作者: hehe123 时间: 2004-11-27 12:59
Exercise ' \) t3 e+ t8 \' R4 T- u) _

Ⅰ. Answer the following questions :

1. What are the necessary conditions for operation to become an object of scientific approach?

2. Point out the main points the 2^nd and the 3^rd definitions emphasize as compared with the first definition.

Ⅱ. 1. Translate the third definitions of OR due to S. Beer.

2. Translate the following sentences into Chinese ;

ⅰ) It was G. Gantor who first introduced the concept of the set as object of mathematical study.

ⅱ) The definition of probability due to Laplace provoked a great deal of argument when it was applied;

ⅲ) Nowadays, we usually adopted measure theoretic foundations of probability initiated by A. N. Kolomogorov.

作者: hehe123 时间: 2004-11-27 13:00
标题: 数学专业英语[14]－The Theory of Graphs
数学专业英语－The Theory of Graphs/ H" D3 q5 H% c- W" Y# b: c9 M) Y9 S2 Y# ~

In this chapter, we shall introduce the concept of a graph and show that graphs can be defined by square matrices and versa.

1. Introduction

Graph theory is a rapidly growing branch of mathematics. The graphs discussed in this chapter are not the same as the graphs of functions that we studied previously, but a totally different kind.

Like many of the important discoveries and new areas of learning, graph theory also grew out of an interesting physical problem, the so-called Konigsberg bridge problem. (this problem is discussed in Section 2) The outstanding Swiss mathematician, Leonhard Euler (1707-1783) solved the problem in 1736, thus laying the foundation for this branch of mathematics. Accordingly, Euler is called the father of graph theory.

Gustay Robert Kirchoff (1824-1887), a German physicist, applied graph theory in his study of electrical networks. In1847, he used graphs to solve systems of linear equations arising from electrical networks, thus developing an important class of graphs called trees.

In 1857, Arthur Caylcy discovered trees while working on differential equations. Later, he used graphs in his study of isomers of saturated hydrocarbons.

Camille Jordan (1838-1922), a French mathematician, William Rowan Hamilton, and Oystein Ore and Frank Harary, two American mathematicians, are also known for their outstanding contributions to graph theory.

Graph theory has important applications in chemistry, genetics, management science, Markov chains, physics, psychology, and sociology.

Throughout this chapter, you will find a very close relationship between graphs and matrices.

2. The Konigsberg Bridge Problem

The Russian city of Konigsberg (now Kaliningrad, Russia) lies on the Pregel River.(See Fig.1) It consists of banks A and D of the river and the two islands B and C. There are seven bridges linking the four parts of the city.

Residents of the city used to take evening walks from one section of the city to another and go over some of these bridges. This, naturally, suggested the following interesting problem: can one walk through the city crossing each bridge exactly once? The problem sounds simple, doesn’t it?You might want to try a few paths before going any further. After all, by the fundamental counting principle, the number of possible paths cannot exceed 7!=5040. Nonetheless, it would be time consuming to look at each of them to find one that works.

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Fig .1 The city of Konigsberg

In 1736, Euler proved that no such walk is possible. In fact, he proved a far more general theorem, of which the Konigsberg bridge problem is a special case.

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Fig .2 A mathematical model for the Konigsberg bridge problem

Let us construct a mathematical model for this problem.rcplace each area of the city by a point in a plane. The points A, B, C,and D denote the areas they represent and are called vertices. The arcs or lines joining these points represent the represent the respective bridges. (See图２)They are called edges. The Konigsberg bridge problem can now be stated as follows: Is it possible to trace this figure without lifting your pencil from paper or going over the same edge twice? Euler proved that a figure like this can be traced without lifting pencil and without traversing the same edge twice if and only if it has no more than weo vertices with an odd number of edges joining them. Observe that more than two vertices in the figure have an odd number of edges connecting them-----in fact,they all do.

作者: hehe123 时间: 2004-11-27 13:00

1. Graphs

Let us return to the example Friendly Airlines flies to the five cities, Boston (B), Chicago (C), Detroit (D), Eden (E), and Fairyland (F) as follows: it has direct daily flights from city B to cities C, D, and F, from C to B, D, and E; from D to B, C, and F, from E to C, and from F to B and D. This information, though it sounds complicated, can be conveniently represented geometrically, as in 图3. Each city is represented by a heavy dot in the figure; an arc or a line segment between two dots indicates that there is a direct flight between these cities.

What does this figure have in common with 图２? Both consist of points (denoted by thick dots ) connected by arcs or line segments. Such a figure is called a graph. The points are the vertices of the graph; the arcs and line segments are its edges. More generally, we make the following definition:

A graph consists of a finite set of points, together with arcs or line segments connecting some of them. These points are called the vertices of the graph; the arcs and line segments are called the edges og the graph. The vertices of graph are usually denoted by the letters A, B, C, and so on. An edge joining the vertices A and B is denoted by AB or A-B.

Fig .3

图２and 图3 are graphs. Other graphs are shown in 图4. The graph in图２ has four vertices A, B, C, and D, and seven edges AB, AB, AC, BC, BD, CD, and BD. For the graph in图4b, there are four vertices, A, B, C, and D, but only two edges AD and CD. Consider the graph in图4c, it contains an edge emanating from and terminating at the same vertex A. Such an edge is called a loop. The graph in图4d contains two edges between the vertices A and C and a loop at the vertex C.

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The number of edges meeting at a vertex A is called the valence or degree of the vertex, denoted by v(A). For the graph in图4b, we have v(A)=1, v(B)=0, v(C)=1, and v(D)=2. In图4b, we have v(A)=3, v(B)=2, and v(C)=4.

A graph can conveniently be described by using a square matrix in which the entry that belong to the row headed by X and the column by Y gives the number of edges from vertex X to vertex Y. This matrix is called the matrix representation of the graph; it is usually denoted by the letter M.

The matrix representation of the graph for the Konigsberg problem is

Clearly the sum of the entries in each row gives the valence of the corresponding vertex. We have v(A)=3, v(B)=5, v(C)=3, as we would expect.

Conversely, every symmetric square matrix with nonnegative integral entries can be considered the matrix representation of some graph. For example, consider the matrix

A B C D

Clearly, this is the matrix representation of the graph in 图5.

作者: hehe123 时间: 2004-11-27 13:01

Vocabulary / j7 Z0 @ h3 j v8 f4 Z6 ^

Network 网络

Electrical network 电网络

Isomer 异构体

emanate 出发，引出

Saturated hydrocarbon 饱和炭氢化合物

terminate 终止，终结

Genetics 遗传学

valence 度

Management sciences 管理科学

node 结点

Markov chain 马尔可夫链

interconnection 相互连接

Psychology 心理学 Konigsberg bridge problem 康尼格斯堡

桥问题

Sociology 社会学

Line-segment 线段

作者: hehe123 时间: 2004-11-27 13:01

Notes 1 j4 a% H" a; q+ ^. p! a" d1 N' A

1. Camille Jordan, a French mathematician, William Rowan Hamilton and . . .

注意：a French mathematician 是Camille Jordan 的同位语不要误为W.R.Hamilton 是a French mathematician 同位语这里关于W.R.Hamilton 因在本文前几节已作介绍，所以这里没加说明。

２．After all, by the fundamental counting principle, the number of possible paths cannot exceed 7!= 5040. Nonetheless, it would be time consuming to look at each of them to find one that works.

意思是：毕竟，由基本的计算原理知，可能的路径的总数，不会超过5040个。然而逐一地去考察这些路径是否有一条路适合题意，那是太耗费时间了，that works 意思是：“有效”，这里可理解为：“适合题意”。

3.It is possible to trace the figure without lifting your pencil from paper or going the same edge twice?

意思是：是否能够跟踪图形而使你的铅笔不离开纸且不走过同一条边两次呢？这一句在英语上等同于without lifting your pencil from paper and without going over the same edge twice.

1. . . .in fact, they all do.

这里they代表顶点vertices; do 代表have an odd number of edges connecting them.

2. A is called the valence or degree of the vertex, denoted by v(A).

注意denoted 前面的逗号，可使读者不至于误会v(A)是用来记vertex的。这里v(A)是用来记Ａ的Ｖalence.

６. the entry that belongs to the row headed by X and column headed by Y gives the number of edges from vertex X to vertex Y.

意思是：属于Ｘ行，Ｙ列这一项的数字给出了从顶点Ｘ到顶点Ｙ的边数。这里the row headed by X意是冠以Ｘ的行，可简称Ｘ行或等Ｘ行。

作者: hehe123 时间: 2004-11-27 13:01

Exercise * V+ J" s8 w$ u1 F8 m2 e" Q

Ⅰ.answer the following questions:

1. How is the Konigsberg Bridge problem stated?

2. According to Euler’s theorem, why is the answer of the Konigsberh Bridge Problem negative?

Ⅱ.Translate the following passages into Chinese:

When a number of electrical components are connected together, we are said to have an electrical network. The junction between two or more components in a network are called nodes of the network, Each path joining a pair of nodes and through interconnections is best described by a diagram which eliminates all the electrical properties of the components. This graph is obtained by redrawing the circuit of the network with lines replacing the electrical components.

The graph makes clear the existence of a number of closed paths which may be traced along the branches. Such closed paths are called loops. Of the total number of loops of a network, a certain number of independent loops may be chosen. One way of choosing a set of independent loops is as follows:form, from the network, a sub-network by removing branches until no loops remain, although each node is still connected by a single path to another node. Such a structure is called a tree of the network.

Ⅲ.Translate the following sentences into English (in each sentence, make use of the phrase given in bracket):

下面简写The Konigsberg Bridge problem 为Ｋ.B.问题

１．Ｋ.B.问题只不过是尤拉所证明的定理的一个特例。(a special case)

２．从尤拉关于图论的一个定理，即可得Ｋ.B.问题的答案。(follows immediately from.)

３．Ｋ.B.问题的不可能性是尤拉定理的一个直接结果。(a direct consequence of)

作者: hehe123 时间: 2004-11-27 13:02
标题: 数学专业英语[15]－Probability

The mathematics to which our youngsters are exposed at school is. With rare exceptions, based on the classical yes-or-no, right-or-wrong type of logic. It normally doesn’t include one word about probability as a mode of reasoning or as a basis for comparing several alternative conclusions. Geometry, for instance, is strictly devoted to the “if-then” type of reasoning and so to the notion (idea) that any statement is either correct or incorrect.

However, it has been remarked that life is an almost continuous experience of having to draw conclusions from insufficient evidence, and this is what we have to do when we make the trivial decision as to whether or not to carry an umbrella when we leave home for work. This is what a great industry has to do when it decides whether or not to put $50000000 into a new plant abroad. In none of these case and indeed, in practically no other case that you can suggest, can one proceed by saying:” I know that A, B, C, etc. are completely and reliably true, and therefore the inevitable conclusion is~~” For there is another mode of reasoning, which does not say: This statement is correct, and its opposite is completely false.” But which say: There are various alternative possibilities. No one of these is certainly correct and true, and no one certainly incorrect and false. There are varying degrees of plausibility—of probability—for all these alternatives. I can help you understand how these plausibility’s compare; I can also tell you reliable my advice is.”

This is the kind of logic, which is developed in the theory of probability. This theory deals with not two truth-values—correct or false—but with all the in intermediate truth values: almost certainly true, very probably true, possibly true, unlikely, very unlikely, etc. Being a precise quantities theory, it does not use phrases such as those just given, but calculates for any question under study the numerical probability that it is true. If the probability has the value of 1, the answer is an unqualified “yes” or certainty. If it is zero (0), the answer is an unqualified “no” i.e. it is false or impossible. If the probability is a half (0.5), then the chances are even that the question has an affirmative answer. If the probability is tenth (0.1), then the chances are only 1 in 10 that the answer is “yes.”

It is a remarkable fact that one’s intuition is often not very good at csunating answers to probability problems. For ex ample, how many persons must there are at least two persons in the room with the same birthday (born on the same day of the month)? Remembering that there are 356 separate birthdays possible, some persons estimate that there would have to be 50, or even 100, persons in the room to make the odds better than even. The answer, in fact, is that the odds are better than eight to one that at least two will have the same birthday. Let us consider one more example: Everyone is interested in polls, which involve estimating the opinions of a large group (say all those who vote) by determining the opinions of a sample. In statistics the whole group in question is called the “universe” or “population”. Now suppose you want to consult a large enough sample to reflect the whole population with at least 98% precision (accuracy) in 99out of a hundred instances: how large does this very reliable sample have to be? If the population numbers 200 persons, then the sample must include 105 persons, or more than half the whole population. But suppose the population consists of 10,000 persons, or 100,000 persons? In the case of 10,000 persons, or 1000,000 person? In the case of 10,000 persons, a sample, to have the stated reliability, would have to consist of 213 persons: the sample increases by only 108 when the population increases by 9800. And if you add 90000 more to the population, so that it now numbers 100000, you have to add only 4 to the sample. The less credible this seems to you, the more strongly I make the point that it is better to depend on the theory of probability rather than on intuition.

Although the subject started out (began) in the seventeenth century with games of chance such as dice and cards, it soon became clear that it had important applications to other fields of activity. In the eighteenth century Laplace laid the foundations for a theory of errors, and Gauss later develop this into a real working tool for all experimenters and observers. Any measurement or set of measurement is necessarily is necessarily inexact; and it is a matter of the highest importance to know how to take a lot of necessarily discordant data, combine them in the best possible way, and produce in addition some useful estimate of the dependability of the results. Other more modern fields of application are: in life insurance; telephone traffic problems; information and communication theory; game theory, with applications to all forms of competition, including business international politics and war; modern statistical theories, both for the efficient design of experiments and for the interpretation of the results of experiments; decision theories, which aid us in making judgments; probability theories for the process by which we learn, and many more.

----Weaver, W.

作者: hehe123 时间: 2004-11-27 13:03

Vocabulary 7 R5 n- J. d. J& M1 t

Probability 概率论 permutation 置换

Plausibility 似乎合理 binomial coefficient 二次式系数

Affirmative 肯定的 generating function 母函数

Estimate 估计 even 事件

Discordant 不一致的 information and communication theory

Communication theory 通讯理论信息与通讯论

Decision theory 决策论 game theory 对策论，博弈论

作者: hehe123 时间: 2004-11-27 13:03

Notes

1. Geometry, for example, is strictly devoted to the “if—then” type of reasoning and so to the notion (idea) that any statement is either correct or incorrect.

意思是：例如几何学就是严格地属于那种“如果，则”的推理类型，所以它也就属于那种对任何陈述要么是对的要么是不对的概念范围。Is devoted to 意思是：“奉献于”，这里可作：“属于”解，注意在and so to the notion~~中，在前面省去is devoted.

2. However, it has been remarked that life is an almost continuous experience when we leave home for work.

意思是：然而，人们已经注意到，生活就是这样一种几乎不断地需要我们从不充分的证据中去做出结论的经历，这就是对诸如我们离家上班时是否要带雨伞做出定时，我们所需要做的。

3. If the probability has value of 1, the answer is an unqualified “yes” or certainty.

这里unqualified解作：“绝对的”，“十足的”。如 an unqualified certainty (绝对的肯定)； An unqualified success (彻底胜利)。注意 qualified 常解作：“有资格的”，“合格的”。如 a qualified technician (合格的技术员)； qualified examination (资格考试，美国高等学校研究生院的一种考试)。

4． If the probability is a half, then the chances are even that the question has an affirmative answer.

意思是：如果概率是一半的话，那么问题有肯定答案的机会是对等的。注意这里even作“对等”解。

5． The less credible this seems to you, the more strongly I make the point that it is better to depend on the theory of probability rather than on intuition.

意思是：这对你越不可信，我们就要强调这种论点：宁可依靠概率论而不愚信直观，这里make the point that 意思是：“主张；强调；视~~为重要” .

作者: hehe123 时间: 2004-11-27 13:04

Exercise 5 v E5 L) Q" K8 M S# u4 M" z

1. Translate the following passage into Chinese

The origin of the theory of probability goes Bach to the mathematical problems connected with dice throwing that were discusses in letters exchanged by B.Ppascal and P.de Fermat in the 17^th century. These problems were principally concerned with concepts, such as permutations, combinations, and binomial coefficients, whose theory was established about the same time. This elementary theory of probability was later enriched by the work of scholars such as Jacob Bernoulli, A.de Moivre, T.Bayes, L, de Buffon, Danial Bcrnoulli, A, M, Legendre, and J.L. Lagrange. Finally, P.S. Laplace completed the classical theory of probability in his book “Throrie analytique des probabilities” (1812). In this work, Laplace not only systemized also greatly extended previous important results by introducing new methods such as the use of difference equations and generating functions. Since the 19^th century, the theory of probability has been extensively applied to the natural sciences and even to social sciences.

2. Translate the following sentences into Chinese:

1. The term random process is use to describe process that gives rise to one of a number of admitted possible outcomes but which outcome cannot be predicted with any certainty in advance.

2. Tow events A and B in a probability model with sample space and probability function P are said to be independent if

P (A B) =P(A) ·P(B)

3. Describe briefly the kind of logic developed in the theory of Probability.

4. Translate the following sentences into English (make use of the phrase or the phrases in the bracket):

设X=[a b], A X (A X) 是一开集，又设a A 令r=sup{ : [a+ ] A}, 求证a+r …A. (这一部分不用翻译，仅需翻译下下面证明部分)

证明：（1）若论不成六，即是说a+r A，则由于A是开集，存在 >0使得[a+r, a+r+ ] A, 从而[(a,a+r+ ) A, 这与r 的定义矛盾。（~~~would not hold, or~~~were false, or were not true; contrary to）

(2)若a+r A,则由于A是开集，存在 >0使得[a+r,a+r+ ] A，由这推出[a,a+r+ ] A,这是不可能的。故a+r A. (this implies)

(3)若论断是错的，则由于A是开集，存在 >0使得[(a+r,a+r+ ) A,从而[a,a+r+ ] A,这就导至与r是的上确界这一事实相矛盾结论。（leads to contradiction to the that）

作者: hehe123 时间: 2004-11-27 13:05
标题: 数学专业英语[16]－The Role of Mathematics in Economics
数学专业英语－The Role of Mathematics in Economics2 M$ R: l# y5 B! z0 ?

Economics is a mathematical discipline. This assertion may seem strange to the traditional political economist, but mathematical methods were introduced at an early stage (Cournot,1838) in the two-hundred-year history of our subject and have been steadily growing in significance .At the present time and, essentially, since the end of World WarⅡ,mathematical methods have become predominant in American economics. The mathematical approach was originally inspired in Europe and England but it has flowered in America, with no little stimulus from European immigrants. The mathematical approach is steadily gaining favor throughout the world, especially because the younger generation in developing economics is embracing the new methods and because the socialist countries have shed a previous bias against the use of mathematical methods in economics. It is clear that the future development of economics will see continued and increasing use of mathematics, although it would be rash to assume that the future course of economic analysis will be predominantly mathematical as it has been in the last twenty years.

The Economic Problem 9 v" W# }- b& F4 w T' [$ _$ L: Z. N" @4 x ' i! j0 P1 \! O* T0 k& z; k

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A favored definition of economics (Lionel Robbins, 1932) is “…the science which studies human behavior as a relationship between ends and scarce means which have alternative uses.” Whether or not we accept this definition as bracketing all of economics, it is a good starting point for our discussion of the role of mathematics .I might want to sharpen this definition by noting that economists try to select among alternative uses of scarce resources in such a way as to make the most efficient (or lease wasteful) employment of resources to achieve stated ends.

Stated in this way, we see clearly that economics involves optimization, and this is the engine that produces principles of economic analysis. We have either a maximum problem or a minimum problem, which is a compelling reason for the use of mathematics. An abstract economy is viewed as consisting of numerous consuming and producing units, who make optimal decisions about their own economic behaviour, given market prices, and then interact with one another to clear supply and demand in markets to determine prices.

Economic theory usually begins with an analysis of the individual consumer who attempts to maximize his satisfaction, subject to a budget constraint (or to minimize budget outlays for the attainment of any given level of satisfaction).The theory then takes up the analysis of producers who strive to maximize profits, Subject to a technological constraint (or minimize cost for reaching a given output level, subject to a technological constraint). These are the typical optimization problems of economics.

The standard mathematical formulations of these problems are as follows. The consumer problem is to maximize a utility function

of quantities of goods and services consumed, subject to the requirement of living within a fixed income

where are the respective prices of the goods and services consumed. The producer problem is to maximize income minus production costs

where are inputs of factors of production and are their costs, subject to the restraint imposed by a technical production function

of the quantities of goods and services produced and of the production factor inputs .

These two formulations pose the economic problem as the maximization of utility (satisfaction), subject to a budget constraint, and the maximization of profit, subject to a technologicsi constraint.We could also formulate minimum problems that seek minimum production costs for producing a given combination of outputs and the least-cost budget to achieve a given level of utility.

Treatment of optimization problems

The consequences of these maximization or minimization problems have been enormous for economics in building a set of rules of behavior. Nearly all economic truths have some root in these or closely related propositions. The original mathematical attack was quite straightforward. Assume that and are smooth continuous functions (with first and second deriva tives), and optimize according to the rules of the differential calculus, given market prices. The necessary and sufficient conditions for optimization define the well –known demand and supply functions of economics of economics and establish many properties of these functions.

For the problem as I have stated it, these solutions are well established and have been in the literature of economics for more than fifty years. Refined points are made from time to time but the ramifications of this theory were made clear in mathematical treatments by Pareto (1896), Slutsky (1915), Fisher (1892), Hotelling (1932), Frisch (1932), Hicks and Allen (1943), Samuelson (1947).

In the 1930's, and again after World War Ⅱ, these problems received extended mathematical treatment ,The extensions were to optimize over time either continuously or in finite incremental periods and to enlarge the number of side conditions. In stochastic models (i.e., those that incorporate chance), uncertainty about future conditions such as price can be introduced. Also, we can allow for the accumulation of tiny neglected factors that always influence human decisions.

The subjective nature of the utility function, led to analysis of conditions in which the results of optimization would be invariant under transformations of the function and to study of the possibility of deriving a utility function, starting from objective demand functions. The latter problem became known as the integrability problem.

It may be remarked that the early development of mathematical economics followed the steps of physics and engineering. There are many analogies between the classical methods of mathematical economics and the laws of mechanics, thermodynamics, and similar branches of science. In some cases, there was a tendency to draw strict analogies that could hardly be rationalized in terms of economic behavior.

An idea that received much encouragement from J. Von Neumann was that mathematical economics should draw upon different branches of mathematics that were more suited to the peculiar nature of the economic problem and economic variables. It was even suggested that new mathematical methods might be developed that would be tailored to economics .In the sense that mathematicians of the eighteenth and nineteenth centuries developed methods that were suited to the problems of physics, we might hope that modern mathematicians would receive inspiration from problems of economics, and social sciences generally. To some extent, this development has occurred in linear programming and optimization theory for situations in which the ordinary methods of differential calculus do not apply. It is up to the mathematicians themselves, however, to decide the significance of this line of development in modern mathematics.

----------Lawrence R.Klein

作者: hehe123 时间: 2004-11-27 13:05

Vocabulary

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predominant 主导的

economic analysis 经济分析

scarce resource 不充足资源

outlay 开支、费用

income 收入

proposition 命题

smooth 光滑

increment 增量

incremental 增量的

stochastic 随机的

derive 推出

thermodynamics 热动力学

rationalize 合理化

market price 市场价格

supply 供应，供给

demand 需求

budget 预算

budget outlay 预算开支

profit 利益，利润

cost 成本

goods 货物

services 服务

ramification (of the theory ) 理论的）细节

side condition 附属条件

作者: hehe123 时间: 2004-11-27 13:06

Notes

1. 象two-hundred-year ( 两百年)这样的复合词，year 不用复数。例如：Five-year-plan (五年计划)

2. The mathematical approach was originally inspired in Europe and England but it has flowered in America with no little stimulus from European immigrants.

意思是：这种数学方法创于欧洲大陆和英国，但是已经在美洲（美国）开花，当然少不了欧洲移民的激励。这里flower作为动词用；而且 with no little stimulus 是一种肯定语气。

3. Whether or not we accept this definition as bracketing all of economics, it is a good starting point for our discussion of the role of mathematics.

意思是：不管我们是否接受这个定义作为概括所有经济学，它都是我们用讨论数学（用于经济学）作用的一个良好起点，这里bracketing 作为“概括”解.

4. An abstract economy is viewed as … who make optimal decisions about their own economic behaviour, given market prices, and then interact with one another to clear supply and demand in markets to determine prices.

意思是：抽象经济可以看成由许多个消费和生产单位所组成，这些单位（的决策者）就他们自己的经济行为——给出市场价格——作出最优决策，然后相互去特约市场的供需交换，以便确定价格。这里

1） who 可理解为units 的决策人的关系代词

2） given market prices 是economic behaviour的同位语。

3） one another 是指units 之间，而不是指market prices 之间

4） clear 这一词用于商业上其意思是：“卖光，买光，交换，清理”等。

5. New mathematical methods might be developed that would be tailored to economics.

tailor 是“裁缝”的意思，这里作动词用，意思是：“使其适用于经济学”

6. Up to 作“取决于”解。

作者: hehe123 时间: 2004-11-27 13:06
Exercise

Ⅰ .Give an example of a typical optimation problem of Economics so as to show that Economics needs mathematics.

Ⅱ. Translate the following passage into Chinese:

Economic analysis has, in the last twenty years, become predominantly mathematical. This is particularly true in the United States, where doctoral candidates now substitute various courses in mathematics for at least some of the traditional foreign language requirement. Economic problems involving optimal decisions by government and business or stable growth of an economy have analogies in problems of physics and engineering that have long been successfully treated mathematically, But economics has outgrown the days when it merely aped the physical sciences in applying mathematics. The author suggests that in the coming era economics may call forth its own branch of mathematics or provide inspiration for great new mathematical discoveries.

Ⅲ. Translate the following sentences into English (make use of the phrase in bracket and see whether one can be replaced by the other or not):

1. 求在下列限制条件下，函数F(x, y) 的最大值。（Subject to ）

2. 设其中是一集合，是实数集，若满足如下条件：

（Ⅰ） ;(Ⅱ)当且仅当x=y时， ;(Ⅲ)对称性： (Ⅳ)三角不等式：其中 .则称是一距离函数.（Satisfy the following condition(s)）.

3.设是定义在区间Ⅰ的一个连续函数，则在区间Ⅰ是有界闭的假设下，我们可以断言，在Ⅰ上一致连续（Under the assumption (hypothesis); claim）

作者: hehe123 时间: 2004-11-27 13:07
标题: 数学专业英语[17]－Polya’s Craft of Discovery
数学专业英语－Polya’s Craft of Discovery% t2 Q9 U5 j! D) ` `( a

George Polya has a scientific career extending more than seven decades. Abrilliant mathematician who has made fundamental contributions in many fields. Polya has also been a brilliant teacher, a teacher’s teacher and an expositor. Polya believes that there is a craft of discovery. He believes that the ability to discover and the ability to invent can be enchanced by skillful teaching which alerts the student to the principles of discovery and which gives him an opportunity to practise these principles.

In a series of remarkable books of great richness, the first of which was published in 1945. Polya has crystallized these principles of discovery and invention out of his vast experience, and has shared them with us both in precept and in example.These books are a treasure-trove of strategy, know-how, rules of thumb, good advice, anecdote, mathematical history, together with problem after problem at all levels and all of unusual mathematical interest. Polya places a global plan for “How to Solve It” in the endpapers of his book of that name:

HOW TO SOLVE IT

First: You have to understand the problem.

Second: Find the connection between the data and the unknown. You may be obliged to consider auxiliary problems if an immediate connection cannot be found. You should obtain eventually a plan of the solution.

Third: Carry out your plan.

Fourth: Examine the solution obtained.

These precepts are then broken down to “molecular” level on the opposite endpaper. There, individual strategies are suggested which might be called into play at appropriate momentsm, such as:

If you cannot solve the proposed problem, look around for an appropriate related problem.

Work backwards

Work forwards

Narrow the condition

Widen the condition

Seek a counter example

Guess and test

Divide and conquer

Change the conceptual mode

Each of these heuristic principles is amplified by numerous appropriate examples.

Subsequent investigators have carried Polya’s ideas forward in a number of ways. A.H.Schoenfeld has made an interesting tabulation of the most frequently used heuristic principles in college-level mathematics. We have appended it here.

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Frequently Used Heuristics : d& B: |* G: s8 ^

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Analysis 4 f( W# v T7 i: w

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1) Draw a diagram if at all possible

2) Examine special cases:

a) Choose special values to exemplify the problem and get a “feel” for it.

b) Examine limiting cases to explore the range of possibilities

c) Set any integer parameters equal to 1,2,3,…,in sequence, and look for an inductive pattern.

3) Try to simplify the problem by

a) exploiting symmetry, or

b) “Without Loss of Generality” arguments (including scaling)

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Exploration . ~' V8 j( T: w- G# r

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1) Consider essentially equivalent problems:

a) Replacing conditions by equivalent ones.

b) Re-combining the elements of the problem in different ways.

c) Introduce auxiliary elements.

d) Re-formulate the problem by

I) change of perspective or notation

II) considering argument by contradiction or contrapositive

III) assuming you have a solution , and determining its properties

2) Consider slightly modified problems:

a) Choose subgoals (obtain partial fulfillment of the conditions)

b) Relax a condition and then try to re-impose it .

c) Decompose the domain of the problem and work on it case by case .

3) Consider broadly modified problems:

a) Construct an analogous problem with fewer variables .

b) Hold all but one variable fixed to determine that variable’s impact .

c) Try to exploit any related problems which have similar

I) form

II) “givens”

III) conclusions

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Remember: when dealing with easier related problems , you should try to exploit both the RESULT and the METHOD OF SOLUTION on the given problem .

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Verifying your solution ) O6 a- y/ ]0 a3 N6 X- K8 t

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1) Does your solution pass these specific tests:

a) Does it use all the pertinent data?

b) Does it conform to reasonable estimates or predictions?

c) Does it withstand tests of symmetry, dimension analysis , or scaling?

2) Does it pass these general tests?

a) Can it be obtained differently?

b) Can it be sudstantiated by special cases?

c) Can it be reduced to known results?

d) Can it be used to generate something you know?

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作者: hehe123 时间: 2004-11-27 13:08

Vocabulary ! ~7 x: f, G' y/ f' e6 z3 }

craft 技巧

enchance 增强

alert 警觉，机警

precept 箴言，格言

treasure trove 宝藏

anecdote 轶事，趣闻

auxiliary 辅助的

appropriate 适当的

heuristic 启发式的

amplified 扩大，详述

append 附加，追加

exploration 探查，细查

perspective 透视

contrapositive 对换的

relax 放松

decompose 分解

pertinent 适当的

substantiate 证实，证明

作者: hehe123 时间: 2004-11-27 13:08

Notes / w" A9 ]. b. b6 K* f1 Z

1．A brilliant mathematician who has made fundamentral contributions in many fields,Polya has also been a brilliant teacher, a teacher’s teacher, and an expositor.

意思是：Polya，一个在许多领域中都作出重要贡献的数学家，也是一位出色的教师，教师的教师和评注家。这里Polya是a brilliant mathematician 的同位语

2．…which alerts the student to the principles of discoveries…

这里alert的意思是：“使机警，使注意”。因此，本句意思是：这种熟练（有技巧的）的教学可使学生机敏地注意到这些发现原则……

3．Polya has crystallized these principles of discoveries out of his vast experience,…

意思是：Polya从他的浩瀚的经验中，把这些发现原则提炼得更加具体而明朗。

4．Rules of thumb以经验为基础的规则，方法。

5．There,individual strategies are suggested, which might be called into play at appropriate moments,such as…

意思是：在那里，提供了许多个别的策略，它们在适当的时刻就会发挥作用，例如……这里call into play意思是：“发挥作用”。

作者: hehe123 时间: 2004-11-27 13:10

Exercise 6 N- V4 B; V# D/ u0 b* z

I．Translate the following sentences into Chinese ( pay attention to the phrases underlined:

1. Note that a+ib=c+id means a=c and b=d

2. We recall that log z: C－{0} C is an inverse for when is restricted to a strip

3. Notice that if ,angles need not be preserved.

4. To show that the test fails when ,observe that, by elementary analysis, and but diverges while converges.

5. To prove the results of this section, we shall use the techniques developed in the last section.

6. We can deduce, in a way similar to the way we deduced theorem A, the following theorem.

7. We are now in a position to draw important consequences from Cauchy’s theorem.

8. We are now in a position to prove easily an otherwise difficult theorem stating that any polynomial of degree n has a root.

9. Unless otherwise specified (stated), curves will always be assumed to be continuous and piecewise differentiable.

10. We shall prove a theorem that appears to be elementary and that the student has, in the past, taken for granted.

11. The solution to this differential equation is unique up to the addition of a constant.

12. The function that maps the simply connected domain onto the unit disc is unique up to a Mobius transformation.

II．Translate the following passages into Chinese:

1． If we do not succeed in solving a mathematical problem, the reason frequently consists in our failure to recognize the more general standpoint from which the problem before us appears only as a single link in a chain of related problems. After finding this standing point, not only is this problem frequently more accessible to our investigation ,but at the same time we come into possession of a method which is applicable also to related problems.

2． In dealing with mathematical problems, specialization plays, as I believe, a still more important part than generalization. Perhaps in most cases where we seek in vain the answer to a question, the cause of the failure lies in the fact that problems simpler and easier than the one in hand have been either not at all or incompletely solved. All depends then, on finding out these easier problems, and on solving them by means of methods as perfect as possible.

作者: hehe123 时间: 2004-11-27 13:11
标题: 数学专业英语[18]－How to Write Mathematics?

数学专业英语－How to Write Mathematics? 5 h1 U* H6 M! g* C, h+ v* w3 o

How to Write Mathematics?

------ Honesty is the Best Policy

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The purpose of using good mathematical language is, of course, to make the understanding of the subject easy for the reader, and perhaps even pleasant. The style should be good not in the sense of flashy brilliance, but good in the sense of perfect unobtrusiveness. The purpose is to smooth the reader’s wanted, not pedantry; understanding, not fuss.

The emphasis in the preceding paragraph, while perhaps necessary, might seem to point in an undesirable direction, and I hasten to correct a possible misinterpretation. While avoiding pedantry and fuss, I do not want to avoid rigor and precision; I believe that these aims are reconcilable. I do not mean to advise a young author to be very so slightly but very very cleverly dishonest and to gloss over difficulties. Sometimes, for instance, there may be no better way to get a result than a cumbersome computation. In that case it is the author’s duty to carry it out, in public; the he can do to alleviate it is to extend his sympathy to the reader by some phrase such as “unfortunately the only known proof is the following cumbersome computation.”

Here is the sort of the thing I mean by less than complete honesty. At a certain point, having proudly proved a proposition P, you feel moved to say: “Note, however, that p does not imply q”, and then, thinking that you’ve done a good expository job, go happily on to other things. Your motives may be perfectly pure, but the reader may feel cheated just the same. If he knew all about the subject, he wouldn’t be reading you; for him the nonimplication is, quite likely, unsupported. Is it obvious? (Say so.) Will a counterexample be supplied later? (Promise it now.) Is it a standard present purposes irrelevant part of the literature? (Give a reference.) Or, horrible dictum, do you merely mean that you have tried to derive q from p, you failed, and you don’t in fact know whether p implies q? (Confess immediately.) any event: take the reader into your confidence.

There is nothing wrong with often derided “obvious” and “easy to see”, but there are certain minimal rules to their use. Surely when you wrote that something was obvious, you thought it was. When, a month, or two months, or six months later, you picked up the manuscript and re-read it, did you still think that something was obvious? (A few months’ ripening always improves manuscripts.) When you explained it to a friend, or to a seminar, was the something at issue accepted as obvious? (Or did someone question it and subside, muttering, when you reassured him? Did your assurance demonstration or intimidation?) the obvious answers to these rhetorical questions are among the rules that should control the use of “ obvious”. There is the most frequent source of mathematical error: make that the “ obvious” is true.

It should go without saying that you are not setting out to hide facts from the reader: you are writing to uncover them. What I am saying now is that you should not hide the status of your statements and your attitude toward them either. Whenever you tell him something, tell him where it stands: this has been proved, that hasn’t, this will be proved, that won’t. Emphasize the important and minimize the trivial. The reason saying that they are obvious is to put them in proper perspecti e for the uninitiated. Even if your saying so makes an occasional reader angry at you, a good purpose is served by your telling him how you view the matter. But, of course, you must obey the rules. Don’t let the reader down; he wants to believe in you. Pretentiousness, bluff, and concealment may not get caught out immediately, but most readers will soon sense that there is something wrong, and they will blame neither the facts nor themselves, but quite properly, the author. Complete honesty makes for greatest clarity.

---------Paul R.Haqlmos

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作者: hehe123 时间: 2004-11-27 13:11

vocabulary 9 y$ G1 s& ?9 `/ P! S, F

flashy 一闪的 counter-example 反例

unobtrusiveness 谦虚 dictum 断言；格言

forestall 阻止，先下手 deride嘲弄

anticipate 预见 subside沉静

pedantry 迂腐；卖弄学问 mutter出怨言，喃喃自语

fuss 小题大做 intimidation威下

reconcilable 使一致的 rhetorical合符修辞学的

gloss 掩饰 pretentiousness自命不凡

alleviate 减轻，缓和 bluff 欺骗

implication 包含，含意 concealment隐匿

作者: hehe123 时间: 2004-11-27 13:12

notes

1. 本课文选自美国数学学会出版的小册子How to write mathematics 中Paul R.Halmos. 的文章第9节

2. The purpose is smooth the reader’ way, to anticipates his difficulties and to forestall them. Clarity is what’s wanted, not pedantry; understanding, not fuss.

意思是：目的是为读者扫清阅读上的障碍，即预先设想读者会遇到什么困难，并力求避免出现这类困难。我们需要的是清晰明了，而不是故弄玄虚。这里fuss的意思是 “小题大做”。Understanding后面省去is what’s wanted以避免重复。

3. I do not mean to advise a young author to be ever so slightly but very very cleverly dishonest and to gloss over difficulties.

意思是：我的意思是青年的作者绝不可有哪怕只是些少，当却是掩饰得非常巧妙得虚伪，我也劝告他们不要去掩饰困难。

4．Here is the sort of thing I mean by than complete honesty.

意思是：这就是我所认为的不够完全诚实的那类事情（东西）。注意：Here is 的意思是：“这里就是---”，然后把要说的事情在随后给出，若用This is the sort of thing---一般是当你把要说的事情已经说了然后用指示代词This来概括所说的事，注意这一区分。

5．In any event: take the reader into your confidence.

意思是：在任何情况，要敢于对读者讲出真相。这里take---into one’s confidence意思是：“对---吐露秘密；把---当成心腹朋友”。

6．Don’t let the reader down..

意思是：不要使读者丧气。这里down是形容词。

7．Complete honesty makes for greatest clarity.

意思是：彻底的诚实就是最大的明嘹。 Make for 是“有助于”的意思。这样简洁而又充满哲理的句子还有 Emphasize the important minimize the trivial.

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作者: hehe123 时间: 2004-11-27 13:13

Exercise + z6 d3 q7 j9 P% {# k) b' q

(Miscellaneous Exercises)

Ⅰ.Fill in each blank with a suitable word.

1. is called the ________ ________of .

2.p (x)= is a_________of_________n.

3. called a __________ _________.

4. is called the________of .

5.The graph of (p>0) is a___________.

6. then is called a________ _________sequence.

7. is a_________of two equations with three_______.

8.Numbers such as and πare called________numbers.

9.The relation between the celements of a set of real numbers denoted

by<(or<;>;>) is called an_________relation.

10.The relation between sets, denoted by is called an_________relation.

Ⅱ.Each ofthe following sentences is grammatically wrong. Correct these sentences.

1. Let is a continuous function defined on[a,b].

2. Differentiating both sides of with respect to x, the equation becomes y’=—x/y

3. Take the derivatives of both sides of the equation ,we get x+yy’=0.

4. The primtive of here C is a constant.

5. We say that has a limit A at if approaches to A when X tends to .

Ⅲ.Translate the following sentences into Chinese (pay attention to the phrases underlined):

1. We are now in a position to prove the main theorem.

2. An analogous argument gives a proof of the corresponding theorem for decreasing functions.

3. An immediate consequence of Bolzano’s theorem is the intermediate-value theorem for continuous functions.

4. We claim that has no real solution, In fact if is a real solution, then we have which is impossible.

5. It is clear that the method described above also applies to the general case.

6. It is easy to show that has derivatives up to order n at the point x=0,where n>1.

Ⅳ.Translate the following passage into Chinese:

1. It is helpful to introduce the words”local”and“global”to contrast two types of situations that frequently arise. If we are considering a given set D, then we say that any specific property holds“locally”at of D if it is tre at if and at all points near ；thus there will be an open ball B apout and the property will hold for all .On the other hand, a property that holds at all points in D is said to hold “globally”in D.

2. The study of sequences is concerned primarily with the following type of question:if each term of a sequence has a certain property, such as continuity, differentiability or integrability, to what extend is this property transferred to the limit function?

作者: hehe123 时间: 2004-11-27 13:25
标题: 数学专业英语[19]－How to Organize a paper (For Beginers)?
数学专业英语－How to Organize a paper (For Beginers)? 1 ^) y. ` V5 U9 I" h7 T

The usual journal article is aimed at experts and near-experts, who are the people most likely to read it. Your purpose should be say quickly what you have done is good, and why it works. Avoid lengthy summaries of known results, and minimize the preliminaries to the statements of your main results. There are many good ways of organizing a paper which can be learned by studying papers of the better expositors. The following suggestions describe a standard acceptable style.

Choose a title which helps the reader place in the body of mathematics. A useless title: Concerning some applications of a theorem of J. Doe. A. good title contains several well-known key words, e. g. Algebraic solutions of linear partial differential equations. Make the title as informative as possible; but avoid redundancy, and eschew the medieval practice of letting the title serve as an inflated advertisement. A title of more than ten or twelve words is likely to be miscopied, misquoted, distorted, and cursed.

The first paragraph of the introduction should be comprehensible to any mathematician, and it should pinpoint the location of the subject matter. The main purpose of the introduction is to present a rough statement of the principal results; include this statement as soon as it is feasible to do so, although it is sometimes well to set the stage with a preliminary paragraph. The remainder of the introduction can discuss the connections with other results.

It is sometimes useful to follow the introduction with a brief section that establishes notation and refers to standard sources for basic concepts and results. Normally this section should be less than a page in length. Some authors weave this information unobtrusively into their introductions, avoiding thereby a dull section.

The section following the introduction should contain the statement of one or more principal results. The rule that the statement of a theorem should precede its proof a triviality. A reader wants to know the objective of the paper, as well as the relevance of each section, as it is being read. In the case of a major theorem whose proof is long, its statement can be followed by an outline of proof with references to subsequent sections for proofs of the various parts.

Strive for proofs that are conceptual rather than computational. For an example of the difference, see A Mathematician’s Miscellany by J.E.Littlewood, in which the contrast between barbaric and civilized proofs is beautifully and amusingly portrayed. To achieve conceptual proofs, it is often helpful for an author to adopt an initial attitude such as one would take when communicating mathematics orally (as when walking with a friend). Decide how to state results with a minimum of symbols and how to express the ideas of the proof without computations. Then add to this framework the details needed to clinch the results.

Omit any computation which is routine (i.e. does not depend on unexpected tricks). Merely indicate the starting point, describe the procedure, and state the outcome.

It is good research practice to analyze an argument by breaking it into a succession of lemmas, each stated with maximum generality. It is usually bad practice to try to publish such an analysis, since it is likely to be long and uninteresting. The reader wants to see the path-not examine it with a microscope. A part of the argument is worth isolating as a lemma if it is used at least twice later on.

The rudiments of grammar are important. The few lines written on the blackboard during an hour’s lecture are augmented by spoken commentary, and aat the end of the day they are washed away by a merciful janitor. Since the published paper will forever speak for its author without benefit of the cleansing sponge, careful attention to sentence structure is worthwhile. Each author must develop a suitable individual style; a few general suggestions are nevertheless appropriate.

The barbarism called the dangling participle has recently become more prevalent, but not less loathsome. “Differentiating both sides with respect to x, the equation becomes---”is wrong, because “the equation” cannot be the subject that does the differentiation. Write instead “differentiating both sides with respect to x, we get the equation---,” or “Differentiation of both sides with respect to x leads to the equation---”

Although the notion has gained some currency, it is absurd to claim that informal “we” has no proper place in mathematical exposition. Strict formality is appropriate in the statement of a theorem, and casual chatting should indeed be banished from those parts of a paper which will be printed in italics. But fifteen consecutive pages of formality are altogether foreign to the spirit of the twentieth century, and nearly all authors who try to sustain an impersonal dignified text of such length succeed merely in erecting elaborate monuments to slumsiness.

A sentence of the form “if P,Q” can be understood. However “if P,Q,R,S,T” is not so good, even if it can be deduced from the context that the third comma is the one that serves the role of “then.” The reader is looking at the paper to learn something, not with a desire for mental calisthenics.

作者: hehe123 时间: 2004-11-27 13:26
Vocabulary

preliminary 序,小引(名)开端的,最初的(形)

eschew 避免

medieval 中古的，中世纪的

inflated 夸张的

comprehensible 可领悟的，可了解的

pinpoint 准确指出（位置）

weave 插入，嵌入

unobtrusivcly 无妨碍地

triviality 平凡琐事

barbarism 野蛮，未开化

portray 写真，描写

clinch 使终结

rudiment 初步，基础

commentary 注解，说明

janitor 看守房屋者

sponge 海绵

dangling participle 不连结分词

prevalent 流行的，盛行

loathsome 可恶地

absurd 荒谬的

banish 排除

sustain 维持，继续

slumsiness 粗俗，笨拙

monument 纪念碑

calisthenics 柔软体操，健美体操

作者: hehe123 时间: 2004-11-27 13:27

notes 3 o/ z/ f$ J% m4 Y7 z

1. 本课文选自美国数学会出版的小册子A mamual for authors of mathematical paper的一节，本文对准备投寄英文稿件的读者值得一读。

2. Choose a title which helps the reader place in the body of mathematics.

意思是：选择一个可帮助读者进入数学核心的标题。

3． For an example of the difference, ……in which the contrast between barbaric and civilized proofs is beautifully and amusingly portrayed.

意思是：作为这种差别的一个例子，可参看J.E.Littlewood A mathematician’s Miscellany一文，在那里，他把野蛮的（令人讨厌的）证明与文明的证明这两者之间的对比很漂亮地和有趣地给予描绘出来，这里“差别”是指conceptual proof 与computation proof 差别。Portray的意思是：“人像”，这里作动词用，作“描绘”解。

4． The reader wants to see the path--------not to examine it with a microscope.

意思是：读者想知道的是有关论证的途径——而不想使用显微镜去观察。这里作者所要表达的意思是：写文章的人只需把论证的要点写出即可，无需把论证的整个分析过程写得过于冗长。

5． The barbarism called the dangling participles had recently become prevalent, but not less loathsome.

意思是：一种称为“不连结分词”的句子，最近变得盛行起来，但这类句子毕竟是令人讨厌的。关于“不连结分词”，请参看ⅡA第三课注2。

6． The reader is looking at the paper to learn something, not with a desire for mental calisthenics.

意思是：读者阅读文章是为了学到一点东西，而不是抱着一种做智力体操的愿望去阅读的。这里作者是在批评有些写文章的人使用了一些令读者摸不着头脑，而要读者去猜其真实意思的句子（例如用”if P,Q,R,S,T”表达”if P,Q,R then S,T这样的句子。）

作者: hehe123 时间: 2004-11-27 13:27

Exercise , J1 Y6 j- V, i! h7 `/ ~5 R5 P

(Miscellaneous exercises (continued))

Ⅰ.Translate the following sentensces into English:

1. 若行列式中有两行成比例，则行列式为零。

2. 矩阵和矩阵是两种常用的矩阵

3. 两个次多项式上的值相等，则

4. 对任意两个多项式一定存在有多项式或者相等于零或者它的次数小于的次数。

5. 如果取双曲线的渐近线做为坐标轴，则双曲线方程将得到特别简单的形式。

6. 抛物线与椭圆和双曲线不同，它没有中心，它的另一个特殊性是它仅有一个焦点。

7. 通过平面上任何5个不同的点，其中没有4点同在一直线上（共线），有一条仅有一条二级曲线。

8. 当椭圆的长轴等于它的短轴时，它化为一圆。

9. 显然，无界序列不收敛。

10. 设在[a,b]上连续，若存在两个点，则必有

11. 若我们能证明连续函数级数在紧集D上一致收敛，则在此集上可对级数进行逐项积分。

12. 此定理给出了用n次泰勒多项式来近似代替时余项太小的一种估计。

13. 用同样的方法，我们还可以证明定理A。

14. 定理中的条件是缺一不可的。

15. 最后，我们再举出两个能说明问题的例子来结束文章。

Ⅱ.Translate the following sentences into Chinese(Pay attention to the words underlined):

1. The compact and Fredholm operators lately have been receiving renewed attention because of the applications to integral operators and partial differential elliptic operators.

2. We dcnote by [x] the greatest integral part which is less than or equal to x.

3. Global analysis on manifolds has come into its owm, both in its integral and differential aspects. It is therefore desirable to integrate manifolds in analysis courses.

Ⅲ.Translate the following passages into Chinese:

1. The concept of ordering was abstracted form various relations, such as the inequality relation between real numbers and the inclusion relation between sets. Suppose that we are given a set X={x, y, z ,…}，the relation between the elements of X, denoted by<or other symbols, is called an ordering (partially ordering, semi—ordering, order relation or simply order),if the following three laws hold (i) the reflexive law, x<x; (ii) the anti—symmetric law, x<y and y<x imply x=y and (iii) the transitive law, x<y and y<z imply x<z.

2. Suppose we are given a relation R (usually denoted by the symbol ) between elements of a set X such that for any elements x and y of X, either xRy orits negation holds The relation is called an equivalence relation (on X)if it satisfies the following three conditions:

(i) xRx

(ii) xRy implies yRx,

(iii) xRy and yRz imply xRz.

Conditions (i),(ii),and(iii)are called the reflexive, symmetric and transitive laws respectively. Together they are called equivalence properties. The relations of congruence and similarity between figures are equivalence relations.

作者: hehe123 时间: 2004-11-27 13:29
标题: 数学专业英语[20]－Mathematical Discovery
数学专业英语－Mathematical Discovery # N7 O: z/ o7 X% X; X

To give the flavor of Polya’s thinking and writing in a very beautiful but subtle case , a case that involve a change in the conceptual mode , I shall quote at length from his Mathematical Discovery (vol.II , pp.54 ff):

EXAMPLE I take the liberty a little experiment with the reader , I shall state a simple but not too commonplace theorem of geometry , and then I shall try to reconstruct the sequence of idoas that led to its proof . I shall proceed slowly , very slowly , revealing one clue after the other , and revealing each gradually . I think that before I have finished the whole story , the reader will seize the main idea (unless there is some special hampering circumstance ) . But this main idea is rather unexpected , and so the reader may experience the pleasure of a little discovery .

A.If three circles having the same radius pass through a point , the circle through their other three points of intersection also has the same radius .

Fig.1 Three circles through one point.

' q9 f9 B4 t4 d' ]" M! v' M$ }8 H " m3 d. c2 y. _! R% e6 O6 @ y) |0 ]# v$ S' J5 X) O

& G: I5 h% H7 I, O

This is the theorem that we have to prove . The statement is short and clear , but does not show the details distinctly enough . If we draw a figure (Fig .1) and introduce suitable notation , we arrive at the following more explicit restatement :

B . Three circles k , l , m have the same radius r and pass through the same point O . Moreover , l and m intersect in the point A , m and k in B , k and l in C . Then the circle e through A , B , C has also the radius

Fig .2 too crowded .

Fig .1 exhibits the four circles k , l , m , and e and their four points of intersection A, B , C , and O . The figure apt to be unsatisfactory , however , for it is not simple , and it is still incomplete ; something seems to be missing ; we failed to take into account something essential , it seems .

We are dialing with circles . What is a circle ? A circle is determined by center and radius ; all its points have the same distance , measured by the length of the radius , from the center . We failed to introduce the common radius r , and so we failed to take into account an essential part of the hypothesis . Let us , therefore , introduce the centers , K of k , L of l , and M of m . Where should we exhibit the radius r ? there seems to be no reason to treat any one of the three given circles k ; l , and m or any one of the three points of intersection A , B , and C better than the others . We are prompted to connect all three centers with all the points of intersection of the respective circle ; K with B , C , and O , and so forth .

The resulting figure (Fig . 2) is disconcertingly crowded . There are so many lines , straight and circular , that we have much trouble old－fashioned magazines . The drawing is ambiguous on purpose ; it presents a certain figure if you look t it in the usual way , but if you turn it to a certain position and look at it in a certain peculiar way , suddenly another figure flashes on you , suggesting some more or less witty comment on the first . Can you recognize in our puzzling figure , overladen with straight and circles , a second figure that makes sense ?

We may hit in a flash on the right figure hidden in our overladen drawing , or we may recognize it gradually . We may be led to it by the effort to solve the proposed problem , or by some secondary , unessential circumstance . For instance , when we are about to redraw our unsatisfactory figure , we may observe that the whole figure is determined by its rectilinear part (Fig . 3) .

This observation seems to be significant . It certainly simplifies the geometric picture , and it possibly improves the logical situation . It leads us to restate our theorem in the following form .

C . If the nine segments

KO , KC , KB ,

LC , LO , LA ,

MB , MA , MO ,

: ~1 J( f( K& N$ j. E& r0 e

$ ^! U, b0 s1 B/ A& |

- X2 V& M1 m# R6 m) u9 n" n+ T

are all equal to r , there exists a point E such that the three segments

EA , EB , EC ,

5 y; `" X0 C. @$ X6 X( U' r

) a: K# t* Q! |3 s3 i

& r5 u+ l* d$ N7 x- {( \

are also equal to r .

Fig . 3 It reminds you －of what ?

This statement directs our attention to Fig . 3 . This figure is attractive ; it reminds us of something familiar . (Of what ?)

Of course , certain quadrilaterals in Fig .3 . such as OLAM have , by hypothesis , four equal sided , they are rhombi , A rhombus I a familiar object ; having recognized it , we can “see “ the figure better . (Of what does the whole figure remind us ?)

Oppositc sides of a rhombus are parallel . Insisting on this remark , we realize that the 9 segments of Fig . 3 . are of three kinds ; segments of the same kind , such as AL , MO , and BK , are parallel to each other . (Of what does the figure remind us now ?)

We should not forget the conclusion that we are required to attain . Let us assume that the conclusion is true . Introducing into the figure the center E or the circle e , and its three radii ending in A , B , and C , we obtain (supposedly ) still more rhombi , still more parallel segments ; see Fig . 4 . (Of what does the whole figure remind us now ?)

Of course , Fig . 4 . is the projection of the 12 edges of a parallele piped having the particularity that the projection of all edges are of equal length .

Fig . 4 of course !

. n, a0 }' y) p* {6 ?4 O* Z* I1 z8 [9 n

$ m: \ u p1 K

: G; n* h3 V/ k6 d

Fig . 3 . is the projection of a “nontransparent “ parallelepiped ; we see only 3 faces , 7 vertices , and 9 edges ; 3 faces , 1 vertex , and 3 edges are invisible in this figure . Fig . 3 is just a part of Fig . 4 . but this part defines the whole figure . If the parallelepiped and the direction of projection are so chosen that the projections of the 9 edges represented in Fig . 3 are all equal to r (as they should be , by hypothesis ) , the projections of the 3 remaining edges must be equal to r . These 3 lines of length r are issued from the projection of the 8^th , the invisible vertex , and this projection E is the center of a circle passing through the points A , B , and C , the radius of which is r .

Our theorem is proved , and proved by a surprising , artistic conception of a plane figure as the projection of a solid . (The proof uses notions of solid geometry . I hope that this is not a treat wrong , but if so it is easily redressed . Now that we can characterize the situation of the center E so simply , it is easy to examine the lengths EA , EB , and EC independently of any solid geometry . Yet we shall not insist on this point here .)

This is very beautiful , but one wonders . Is this the “ light that breaks forth like the morning . “ the flash in which desire is fulfilled ? Or is it merely the wisdom of the Monday morning quarterback ? Do these ideas work out in the classroom ? Followups of attempts to reduce Polya’s program to practical pedagogics are difficult to interpret . There is more to teaching , apparently , than a good idea from a master .

——From Mathematical Experience % W- d4 V: R) Q5 I

0 p$ x" _' `. e" P* H

& m" {4 Y. O+ O8 D6 H! j

) ^& f- u5 j, m7 C. D/ | i9 g) x

) W! m8 S/ w% A3 R# w

作者: hehe123 时间: 2004-11-27 13:31

Vocabulary

/ z |+ u& e5 P, m

subtle 巧妙的，精细的

clue 线索，端倪

hamper 束缚，妨碍

disconcert 使混乱，使狼狈

ambiguous 含糊的，双关的

witty 多智的，有启发的

rhombi 菱形（复数）

rhombus 菱形

parallelepiped 平行六面体

projection 射影

solid geometry 立体几何

pedagogics 教育学，教授法

commonplace 老生常谈；平凡的

作者: hehe123 时间: 2004-11-27 13:37
标题: 数学专业英语[21]－Notations and Abbreviations (I) Learn to understand

数学专业英语－Notations and Abbreviations (I) Learn to understand2 \6 X7 ~9 e& t) O: I' r

9 t3 c. X4 ~1 P% g. _ ]4 @& s

N set of natural numbers $ {; U! u: K2 L6 S, o+ P/ ? $ n$ z* G% e: d- k9 \7 I9 E : _& t" W8 \5 V6 r7 X

% O4 V1 Q, B4 f9 r

Z set of integers 6 `4 O1 f; j; {- }) X1 a. V

- h* b4 q! i( @5 n: U: d8 @

3 O5 ]8 a7 p2 l2 t# p) E! W

R set of real numbers ' y$ @0 Y( H+ P8 g7 @8 s* e

4 b) q9 m e& z6 i9 c8 g' E

# n+ m* e# }5 D X

C set of complex numbers ; r) ~$ z# ]$ i$ `8 W6 b

4 V, s3 J d* m

+ T9 [/ r, c1 V' E. G9 Y

+ plus; positive ) e& k% G e. X: Q2 c: }

; @1 ~) n& E5 ?7 U

8 i+ L( y" v+ D5 C

－　　　　 minus; negative 6 x& k: g: q% }# K* E

* K7 G" l! j% p

+ c& _; @' P% P+ {0 z

×　　　　 multiplied by; times - m3 P$ v7 a# G h+ R1 v1 k

' L$ s* X7 ?/ w1 }3 R' N& L

2 H6 Q0 K% V, I8 b+ {" ]1 S0 g a3 l

÷ divided by 2 v2 F0 z& Z$ k; o& Q

9 \& _- O3 a1 k! D: b

@4 C0 z1 O6 C8 }/ J2 m) h

＝ equals; is equal to ( d* e- ~$ {& i/ C+ O, Z6 E6 S

8 L) y9 o, a, l7 q- b9 [8 n/ ~

: z# H5 D8 x9 i8 w J. L0 M

≡ identically equal to # `7 J; A1 H( `! {: U

2 g8 O' q* Z+ K) k' h- c

{* [+ H9 N2 g

≈,≌ approximately equal to * a b, M, ]5 T7 L/ F/ R" d8 ]+ E

0 z: p- b' x3 B9 E8 Z1 R

* X0 R, C' l1 c0 }1 Y. p+ w

＞ greater than / \, O' `) E4 N

# s4 |1 T7 t! T' Y

# U- Q8 {* u9 ^: C- Y$ |

≥ greater than or equal to # m. o; R8 i& q# N( W- z# Z* T/ F

% s* x$ G& ~ Q7 q

! _) @9 h9 u: g# w( Z8 n- A* r. D

＜ less than , b. ~& M& \" O6 x5 X% p

- V6 J, o, @ n* D

1 E7 C6 C$ M. G1 K* T6 C

≤ less than or equal to 9 }( e5 c+ b) x1 a/ V4 m) W

$ t2 [) w) W, u2 z U

: u+ j. x8 a* p7 ]+ f

》 much greater than 6 q1 f0 T b" O- _7 q/ ?) {

" K7 A$ m3 S$ v: p9 i

. K) \) Z7 l1 o

《 much less than + c1 M) P7 r. `! _

5 ~, \ ~4 e% h, S5 U3 H9 m7 q

$ I5 {- x# t! y

square root 6 z5 C% f$ d& I1 i4 Q4 ?& Q. Z( l

6 [; o. T, }5 Q- t3 q

7 W# l P8 h5 f$ d Y/ o8 T, ?

cube root ( g4 T4 S( \8 l. o3 d

6 u9 m8 i( t4 U P# N

' s3 G; P$ a5 C

nth root ; M- o8 }1 S- r5 S; a

! @% M, }& f0 B* Q: b9 [& c4 J( ]

' w5 A+ y1 L6 o3 v+ K( S

│a│ absolute value of a ) e3 B5 j; Y, z! h( U; I

1 M- [1 {" g, B

, L3 G$ f6 a4 s* S

n! n factorial / z$ S. e, M0 j8 B0 M" h# ^( s. F

1 M2 ]8 O1 {$ K, [+ j1 S

3 Q& h( Q9 N A0 A

a to the power n ; the nth power of a ' w% A( W8 T7 j8 ]. s! f: C2 g g

( P o1 d- ?3 u* N

3 D9 x& _6 R7 O2 t& x! W! {

[a] the greatest integer≤a $ C$ h6 C4 ?4 h% g& w. t2 R

4 {/ O! E, c- _. C& c# w6 |0 l

+ p) o( ~8 n7 v9 M) ]6 T! `5 n

the reciprocal of a 1 V& v+ C- {3 J, F3 t; O

& u' g1 G5 ]( v

* P( E* R8 X% n1 S

/ X6 q; e, |, ]( j7 z4 s9 |, M

$ B* z, O4 k# Q

" U) i6 Y: n' H; _

Let A, B be sets , e; ?3 D" [1 X% y; w3 r9 g4 c

& D$ W( Y" n: X2 V

! x t* E& S* L0 Y( T/ x0 \

∈　　　　 belongs to ; be a member of 1 ]& t; [1 Z6 t8 e

# D7 X$ P( Q$ c3 _/ }* d

/ r. Y! a$ |( _: o7 o: l, I) y" B

not belongs to 8 m! s8 @- s2 y9 R& A( n

, W5 [1 _* ~. Q! N

8 B9 i/ f3 w# F& n% Q; B0 l

x∈A x os amember of A 4 z, Q$ f6 ?5 B+ X

! k. U- H! Z8 D. [0 u! E6 |! O

0 A9 m% o4 t) A/ t% } I

∪ union + e) X* ]- _ c4 i: }5 M0 Z

: \* _/ v, ^8 p( W8 d8 U5 b- U, B& i

. ]5 H( M S' x$ G; }8 }) H/ ]

A∪B A union B ( Q: r) [2 H- {8 w8 w

% f! Z$ ^) W* i/ I# M# x) E

0 O) o$ {5 V3 P

∩ intersection - P6 X" ]# u& w( |! T6 R& r

) h. |# G# S6 V( `( O

8 K" }! _* G& j( _

A∩B A intersection B - ~% X2 U& X/ P, ^, T9 `& g3 p% A

: U5 }5 p1 t# J/ Z4 X! x

# T5 u" A# U9 m8 z* z4 {1 F/ `

A B A is a subset of B;A is contained in B 4 P( w6 p) Q5 Q5 C+ c: n

) i0 X- B! z! [5 [; s; N

3 O! ^* A7 w& n0 P$ H

A B A contains B 6 ^* ^3 C" X9 j6 a9 D; ~7 P0 j

5 L7 w# s8 b i& O# w y1 J; D' b

" m- H( w5 B% r6 F+ P. K( z

complement of A ( L1 u# _# h7 o$ n& {' e

. Z0 G: |. k( j( s9 r

0 H) k+ `- n! `8 `4 l

the closure of A ; y: z2 T8 ?( N$ @8 C `( |& b& G

# F5 q4 n: B! x+ D' D: B

6 \: r. Q3 K$ V7 e

empty set ' M: m* j% v! V3 O0 N& j4 p6 ^

0 b, ?# B2 S& W" H0 |+ a1 [2 t& ]

; h3 I- a+ \ s

( ) i=1,2,…,r j=1,2,…,s r-by-s（r×s）matrix 0 Z5 D; |% u. U7 X

7 z7 d% Q; P% T9 \

( z/ e+ K) X: {) o3 h3 C

│ │I,j=1,2,…,n 　　　　determinant of order n ! N- j$ j7 S$ H1 p' p1 A

' M& J3 m4 {# c: M6 ~& A

9 f! x6 G) ^1 H

det( ) the determinant of the matrix ( ) ' _5 }- R, ]$ z: [4 e9 q

: ~6 x5 P& O4 A

4 W6 |, }: o0 H& {6 [* K

vector F , P6 l$ A1 x+ X. n

; X s; u/ s$ _/ r

5 V7 H6 V: A1 R. R6 m8 c3 i) E$ Q

x=( , ,…, ) x is an n-tuple of - }" Z4 r0 ]) p- F$ C: I

- k0 w3 b' Z* X4 N3 E: T, C5 }

3 d' k- }; H8 H& P5 w

‖‖　　　　　　　　　　the norm of … 4 W# e. T' O @1 M3 A7 u! R

+ d& L! L+ e, r) Z! s/ ?" [- B: F

- U; F- E" Y% @% U6 J% H

‖ parallel to 7 ?; i! _: X( d/ ]2 w9 Q) H3 C

& h' w! w5 m' |4 }1 U

5 D1 q9 g/ H% v* n

┴ perpendicular to / p) \0 R- O6 `# c1 e

A$ L7 T5 S# r% h% c

5 y$ S& _; Z; ]

the exponential function of x ' E3 Y: B5 t: x6 @8 S' b& G

4 w2 E. U" ?' K8 i& B/ W

) u! Z2 `% o8 @6 x8 q

lin x the logarithmic function of x 1 R- k+ u* I2 a M) ~

6 E/ P% @. Z" i7 _- D# Q

2 Y6 g& I9 p" k7 T3 j

sie sine ( U; A# d& t& q# }! U

5 p7 I+ `; T0 f6 L8 I

% k- b$ j; W' A4 k0 i; B. E% @0 d2 I

cos cosine C! G R6 p- A# t* u$ b( b5 b

, f5 J, }, I* r1 M& A. d

0 w/ U4 T: k0 V& X9 t& g

tan tangent 4 S0 ]; b0 c$ W8 Z: q. W& a( C

* Q- v1 [: o9 l m7 w9 p6 L$ y

2 V7 z; w7 n9 ]( R) Q( E0 ~

sinh hyperbolic sine - _1 W. d% T: q3 `* A

4 t, T8 W, n, O1 w2 H, x2 ~

: a0 K$ A. L/ `, S i& X

cosh hyperbolic cosine * I$ g1 {! S+ }6 y4 N- o

5 @% r/ J+ z; L# a: |+ p7 ]

! h! a, k1 }8 C% R/ |- j" h

the inverse of f 2 I+ l- U4 k; c

@: M- U0 d; E X/ K, a w

/ |$ b8 D$ d* r( [* M) W+ a

f is the composite or the composition of u and v & E2 |* _* a) k7 b4 _

% {/ Y6 {. o+ H5 s

+ f# @0 U8 ~+ V% h+ a

the limit of …as n approaches ∞(as x approaches ) ' ]% `! @( r0 {; P

( |1 A! E+ e2 X/ t( v* _

: I4 ?' i' w/ d

x a x approaches a # L! U6 a6 \7 t: H$ r

3 ?' N; b6 s6 \% J# d; ]

; p, {" P' a1 R9 @

, the differential coefficient of y; the 1st derivative of y + Y1 m' b8 x0 P0 h* O' b

& H) U" q! D( b# q& x( |

* Y0 \2 q8 _& X( a9 p

, the nth derivative of y + F- a5 x# a/ _+ ]0 ?+ G

% w: n* I( y) L6 j& n% r# r$ p

$ `! S k" y) u, H% L9 ]$ d

the partial derivative of f with respect to x ; W9 S: b5 F( V& z) {" x) i1 \' ~

% D; a x1 o5 `

/ r7 q; z M" e/ r

the partial derivative of f with respect to y & v& W4 w$ h, c+ W

5 E* ]+ D9 D3 N

+ V T0 E$ p6 u3 _( Q! b

the indefinite integral of f # u! D. ^) L2 A1 ~+ c4 ^( K6 R

, p$ K. S! X' m( A$ g$ {5 Q

+ {, h" h) ]! T9 b

the definite integral of f between a and b (from a to b) + T; l5 j4 {4 r8 ~2 h6 J

( Q% H/ o3 ]& M4 K

3 `8 l! }0 R# T/ n2 v1 ]9 R

the increment of x 8 _+ u4 g& a5 ~& V; ^- M8 f

. l; R: w6 Q* ^% w6 H9 ~

8 t4 w% @/ y+ l' J( r. b1 |

differential x 1 |$ B( g! W- e$ o

$ @& I! x: U* a* _

% [' X, ]7 U8 z% H) p

summation of …the sum of the terms indicated 5 B: `7 E5 r7 j7 w0 _& L- M

8 q4 R$ g d9 X

$ p4 }, P7 l' K: w/ s [) I! ^5 p

∏　　　　　　　　　　the product of the terms indicated & G- [( J, D9 A. {% n# \9 o

: {! u W% x$ a2 @

) t5 z' X5 M5 K3 `5 G3 _

=> implies : J% ~, K" F4 b7 f% P7 o

. P& C9 P) A8 k) Y" Y3 N

X& |! f# F* s- P: {: o

is equivalent to ) ~, F' I: i2 |# V

+ `, L& M4 N+ H' n9 D

6 J: g& Y; ?( u/ }4 ]7 M8 o) K

（） round brackets; parantheses & D/ _; N+ [( U& @+ i; S3 G3 Z

5 z$ \, p) P7 Q

, y# O d) z8 J4 g0 C. l4 A# d

[ ] square brackets ! P7 Z2 q+ @ R% f3 z. `! [

) O! S; X4 T% @% ^0 s. ~

1 F3 S3 v8 Z7 M0 N) p

{ } braces 0 d' q: J$ b. O7 R

9 e' {/ \5 f& h" o) y' m

% R8 H/ K# m5 s+ ?, l0 C) @

' W8 t+ u; f0 J6 e+ M) M' @0 z& b

% L( {$ e* W6 R/ }+ ^" A' L% o

o" j% K; n4 w

& c* ? G+ {3 \

作者: hehe123 时间: 2004-11-27 13:38
标题: 数学专业英语[22]－Notations and Abbreviations (II) Learn to read
数学专业英语－Notations and Abbreviations (II) Learn to read % ], ?8 S( J) `) A3 e9 F" T" X

8 q, N! D% B7 |/ [% x

0.1 0 point one; zero point one; nough point one P9 |( A& n: N1 { ; m# \ g1 B% k/ x! }; B6 w7 N4 V0 K3 m! S

& m6 ^( S Y; u7 D2 _: F. h

0.01 0 point 0 one; zero point zero one / e2 W1 ?( ~3 u( I4 R+ S

1 {+ B* k( e4 Q7 m5 ]

T0 z: T, P; b% i3 k

4.9….. four point nine recurring 7 W# j' f3 g5 v( p5 H: |" c. u

: i( Q/ U$ d# b

4 @! X0 \. Y, E* |/ r

3.03262626… three point nought three two six, two six recurring - Y& t7 `8 K" L# `3 e8 ]

* ?9 n* t. S; z' f9 ?+ a

! l, w" y" c9 Q+ \6 y" d

38.72 thirty eight point seven two 8 J7 w7 Y+ M. i. G+ ~

: D" z5 s: D! V( k( D

7 p0 ?3 d b/ l3 c7 \' f9 C9 u& C

a+b=c a plus b is c 9 W4 O5 p( @/ B9 r) t

; s. _. @, [* a9 G

) `8 ^' `" A2 ]5 @

c-a=b c minus b is a ; b taken from c leaves a : P) h2 t$ n1 H/ Z

2 B; V1 `/ y9 h

6 K9 ~% r4 U' N1 P% [( E

12÷3=4 twelve divided by 3 equals four . ~8 d6 _( r6 w3 V1 ]

5 d2 M- w; @& P9 r2 K9 w# [9 \

! ~$ {. ?9 t& L" B% H

30=6×5 thirty is six times five 7 F: z3 d) N5 M6 Y6 N1 ~

! z3 T4 P2 B5 S' x$ [, I' j; c

" W2 {2 D z, V3 }- e, u" v

6×5=30 six multiplied by five is thirty : w4 m W1 C0 h7 C

+ K" K. [, s8 j- ^, F8 S

! w7 D/ S$ a: A$ m) r6 t5 J

7+3＜12 seven plus three is less than twelve . L2 K6 R) I9 O

! `9 i8 n; x# E' o4 s) m

2 W2 {) u' U/ p( p, U

(a+b) bracket a plus b bracket closed 0 X8 g2 |8 U- m- C. s2 s

8 O6 V$ m3 b' d, x

/ l! U( R8 I+ u" K: w7 k1 p- o

20:5=16:4 the ratio of twenty to five equals the ratio of sixteen to four ) ~7 ] y+ X- k' N

/ S- @+ a7 ?3 c0 k( e2 O, _2 n

- K% S2 G% D" ]2 }$ [7 b# J

a:b∷c:d a is to b as c is to d : y: e' V3 |' H2 [4 z0 i4 J

P3 l3 ^- i: g

& S+ e7 p0 F3 c+ i8 l# t& {0 W

v=s/t v equals s divided by t; v is s over t " K' z+ n% Q1 A2 p5 C

, g- c/ h4 o% S

8 n% }$ ~2 a1 G. C7 f7 \

(a+b-c×d)+e=f a plus b minus c multiplied by d, all divided by e equals f 5 s' H6 z8 b+ Z4 U) P" c

3 n% |/ Q1 V& a! g4 [- l: F

' q, u: p2 s* w# B6 A: Y% v' F

% percent 9 v" O0 u& Z7 ~/ l

8 v, H. i6 f5 h& h; C8 ]5 ?

' L& g4 o# {, r" X

3/8% three-eighths percent , \6 M* K6 ` ]: n6 A) ?7 Y) y- i5 F

" i% u5 S* v& c3 o [8 p- U2 X; F; m

& x1 k; x* H4 m# w$ X

x square; the square of x & D* q' C) l! n! t8 T

& Y- N" N0 {0 X

3 W" H' j. j! q( K: Q4 `

y cube; the cube of y , y/ m& U7 x7 B" u

* W$ d2 f% b9 A" D

$ m, R6 [ ]. }) w* M" z

the cube root of a + M3 m$ k A8 O" b8 A! w0 |

9 f- e& M6 m$ P( I0 _. e' u" z

+ @* {0 e! C' X' _

the square root of five hundred and eighteen . d+ ]* p2 m) B5 r) S( S. X

) M" @) o' N% }. Z1 o

2 k' i2 h% R$ ?0 R

the nth root of the difference of b and c # ]7 v% I& c& w0 x1 [

0 i3 A1 u1 a3 |' i+ g6 j

) U+ N8 { r! H1 \

lin x the natural logarithm of x; the Naperian logarithm of x $ u! q( Z [$ ]- `' f* }

$ D7 K- U0 k3 ~

0 M$ o! v, \* [" u) A/ ?+ X

log x to the base N 4 T- o) a. O% N- m% W" l

: i2 w' Y" L. L+ _

; S5 ]! T$ s7 `1 J. p8 y

sigma : P+ R# _- Z" j1 c

! T1 r1 S- P9 ] B

5 {! e7 q$ u/ l( i

b prime 6 |2 @' o$ N, B; d

7 B% l! U) ?+ n$ o# g% m

( W" d% c' O8 F- D# w

b double prime ) s; \1 j' B: V* I

! E7 u' H4 B$ Q1 L" j+ |

. b) u, \% U% I

b triple prime $ r9 Q, }) t/ @6 f" p1 _

& v% n9 Y: }/ N9 z! e

* J) \! x( n6 M, {

b sub m ( m) R1 J+ o, h

% ]- q8 p: H# X9 R

4 l4 h; Q$ ]* H U$ X0 T5 w

b double prime sub n 1 [, k& `; H0 {. C

C( }. M% v7 ?: q

# y. h s' W# y G/ M. b

……… dots & V" f# x' H& O) d

6 V7 L5 V. G$ A( x; |

; y3 S6 |3 s* k; P% M

π pi : b$ U% H4 n+ b: @1 L

4 Q* Z$ p/ r. i) }0 W* D! u9 X* ~

; }7 e9 m9 I2 X$ B5 v1 I# Y

α alpha 9 v! X0 j' L" y6 \) A

& Y7 w. x7 w0 `4 Z- _# x

6 |1 I5 B/ o* Q; {1 s# t+ t' F1 w

β beta 0 \4 I# h( B8 \9 F9 Z

6 h$ D/ q) Z3 M' K) [

0 d) {" ^7 v4 ^5 ^% {4 X' k" x9 x* G

γ,Γ gamma 9 l1 q% e) F& Z

5 r, F( z j4 w% r6 d7 S

' K: I, h$ k9 }

δΔ delta - f/ B8 l: @) U7 A

9 A2 _+ @1 U/ G

) ^' `/ i; V" T) O

ωΩ omega 7 G% j. k8 \/ T0 g O9 ^

! g( L: h1 D6 D% c8 I2 Q8 o

; j# j' G# E! L5 D' h O

ξ xi ) }& w$ e, `, {. b

! {5 u$ Y; c% U" i% z

& o4 t4 a1 V: `

η eta 8 E% S0 O6 k/ P9 r" \5 m6 K6 c% t

3 k, e' C4 m0 r" A( k) ?0 t6 s

) I" Q- E4 {* @ ^& [# e; }

ζ zeta 8 y' L+ g1 Z/ M. C- ~3 u) l

0 ~1 O3 g% x. }" ?8 G3 ~7 r

5 V# W( j0 H. h! z

Φ phi * p* R; q) O1 h7 N3 N. W* w& M8 {$ ?

- ^) N$ t8 e6 F5 {0 f3 I# v

. p+ e g* i4 K) Y

ψΨ psi 2 ]1 j: L4 w d

1 J& ^% @9 R- Q! x

% _5 @" L0 h4 C8 i1 d% M4 c

χ chi . K/ p3 z" P3 L) d2 H( n

0 J4 Z1 r. V7 w0 |9 A* s, u

* R0 Q4 L+ m' n ^: L

ρ rho , _" h7 z1 Z% V7 `/ H

p6 U8 U+ Z! H& W0 C

' d& N! `" d$ P E

τ tau % p1 _) v" ~) ~& \

* Q Q. g4 V, C+ ~9 d% ?

* _; X* s6 j2 A% X/ y

υ nu ' q3 d. f1 |# {) I- P2 b& ]

2 J7 X+ c5 }# W7 g

8 y! Y8 m0 O# d8 ^; x- c) }

μ mu 6 J) @7 q$ j/ w( t2 m9 C& P

" y5 a+ x( B: r. K

4 T+ o, w5 W; H) J/ g5 d! v+ C

λ lambda / b% X6 ~' t H8 e/ q2 P) G

& q2 N3 ?" o4 e4 d3 S0 B0 T

8 ^' i) Q0 W" t, e; a& t

κ kappa ( x. K" `* V/ E5 ]- k6 v

- N: c* @: M: e; {- }. f" x

2 |- p! v1 S4 O- w5 y) s

ε epsilon 6 K3 N! d! c7 @# A7 U" _6 H; I

( Q0 J; D& }8 L9 K4 e- [8 W

3 ]- n( @2 L. z( C; u7 O* q

θ theta . I& J0 o+ ?* P, R% Z/ W' ?% ^1 m

/ m9 F! z: u4 {/ Y+ o; P

7 K! _, G" n6 d

∴ therefore 7 x3 w( ` B) x

! z% `6 e/ q* K$ P) O5 ^* |; i

& S; f# I. I/ O- C# `

∵ because * c6 t# f. X: M/ g8 l: l* s

3 q* b, n/ y& L( |! k% y

7 v# I; O+ v) E

there exist (s) + v0 W7 Z+ T/ D n1 C7 z- e

0 h# `( W2 T# `

0 q0 x/ X: v$ M: V. u, |" G

for any ) z, s: k8 }4 P4 w' U- \

! X" }0 k# r$ Z9 s

; B9 m: z1 E( N& l D J: |

iff if and only if / {! i0 s0 P, E0 {* o) ~1 u

* f" g; B9 f% d" n) ]

) I& ^7 t N2 w3 D N* b

etc eccetra & Y+ L( M5 M5 S& A( F! ~) t

! O$ {, {* T, R+ `" ]( h i1 K P

" ]. u& S5 h- |$ i- I5 E" U: Q7 U% z

e.g. for example 9 l" r' G* n* N

' V0 d7 ]9 ~8 v* f- @4 h

' i- n7 I+ _1 b

i.e. that is ) X0 N' T" d. V; R

% x4 U4 G8 y! S; B& Z0 y

4 i/ D" G- Y% c/ k9 K8 j# P( W3 p

viz. namely 3 k& [% r! p& x8 r4 ~

" j. `6 m( j# a, {

/ q. c0 ~% |0 H

w.r.t. with respect to 6 _* q! o3 O& l* S8 r8 m' K

- M" y: H9 P. W' P

4 e& [( Y- F' z8 | r/ D8 L- I! d, n

' V5 f( H& T3 J1 }3 ?2 f

; [/ n+ j4 E1 u1 k& p- J7 \

( j+ D; }0 U6 N5 V3 W6 k7 C

7 b2 \1 a1 s' ^ n1 }" Q/ `1 [ u

2 f* X4 \" e5 [# B' L2 p, m- O. A

作者: lipu_2003 时间: 2005-6-14 12:49

好的，谢谢了

[em04]

作者: lipu_2003 时间: 2005-6-14 12:50
好的，谢谢了，你真伟大，有用

作者: lipu_2003 时间: 2005-6-14 12:51

好的，谢谢了，你真伟大，有用

你要是发成文档就更好了

[em04]

作者: jf_102 时间: 2008-1-14 09:15
长知识了

作者: lovecan 时间: 2009-2-3 12:59
谢谢分享！

作者: lovecan 时间: 2009-2-3 13:43
谢谢分享！

作者: xiaoleif 时间: 2009-6-15 10:05
楼主有么有全部的翻译啊？？？？？？？

作者: ykl126 时间: 2009-6-27 22:21
楼主辛苦饿了，这个我很需要的啊！

作者: zhula8626 时间: 2009-7-16 11:01
这太稀少了，分享之

作者: dolphincc 时间: 2009-11-11 21:01
不要钱的资料啊，谢谢无私奉献。。。

作者: 540982016044 时间: 2010-1-15 20:36
不懂。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。

作者: bilinguistic 时间: 2010-1-23 15:34
不错啊！！！！！！！！！！！！！！！！！！！！！！！！！！

作者: bilinguistic 时间: 2010-1-23 15:36
谢谢分享啊！！！！！！！！！！！！！！！！！！！！！！！

作者: bilinguistic 时间: 2010-1-23 15:37
多谢分享！！！！！！！！！！！！！！！

作者: bilinguistic 时间: 2010-1-23 15:38
谢谢分享！！！！！！！！！！！！！！

作者: bilinguistic 时间: 2010-1-23 15:42
多谢了！！！！！！！！！！！！！！！！！！

作者: bilinguistic 时间: 2010-1-23 15:43
多谢了！！！！！！！！！！！！！！！！

作者: bilinguistic 时间: 2010-1-23 15:45
不错
11

111111111111
1111

作者: bilinguistic 时间: 2010-1-23 15:46
多谢！！！！！！！！！！！！！！！！！！！！！

作者: bilinguistic 时间: 2010-1-23 15:47
多谢！！！！！！！！！！！！！！！！！！！！！

作者: bilinguistic 时间: 2010-1-23 15:49
多谢！！！！！！！！！！！！！！！！！！！！！

作者: bilinguistic 时间: 2010-1-23 15:49
多谢！！！！！！！！！！！！！！！！！！！

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